From mathcomp Require Import all_ssreflect.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing]
From hanoi Require Import extra gdist ghanoi ghanoi3.

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Section Hanoi3.

(*****************************************************************************)
(*  The pegs are the three elements of 'I_3                                  *)
(*****************************************************************************)

Implicit Type p : peg 3.

Local Notation "c1 `-->_r c2" := (rmove c1 c2)
    (format "c1  `-->_r  c2", at level 60).
Local Notation "c1 `-->*_r c2" := (connect rmove c1 c2) 
    (format "c1  `-->*_r  c2", at level 60).

(******************************************************************************)
(* Function that builds a path from peg to peg                                *)
(******************************************************************************)

Fixpoint ppeg {n : nat} p1 p2 :=
  if n isn't n1.+1 return seq (configuration 3 n) then [::] else
    let p3 := `p[p1, p2] in
    [seq ↑[i]_p1 | i <- ppeg p1 p3] ++
    [seq ↑[i]_p2 | i <- `c[p3] :: ppeg p3 p2].

Lemma size_ppeg n p1 p2 :
   size (ppeg p1 p2 : seq (configuration 3 n)) = (2^ n).-1.
n:nat
p1, p2:peg 3
size (ppeg p1 p2 : seq (configuration 3 n)) =
(2 ^ n).-1
Proof.
n:nat
p1, p2:peg 3
size (ppeg p1 p2 : seq (configuration 3 n)) =
(2 ^ n).-1 
elim: n p1 p2 => //= n IH p1 p2.
n:nat
IH:forall p1 p2, size (ppeg p1 p2) = (2 ^ n).-1
p1, p2:peg 3
size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
rewrite size_cat /= !size_map !IH  expnS mul2n -addnn.
n:nat
IH:forall p1 p2, size (ppeg p1 p2) = (2 ^ n).-1
p1, p2:peg 3
(2 ^ n).-1 + (2 ^ n).-1.+1 = (2 ^ n + 2 ^ n).-1
by rewrite -(prednK (_ : 0 < 2 ^ n)) // expn_gt0.
Qed.

Lemma last_ppeg n p1 p2 c (cs := ppeg p1 p2 : seq (configuration 3 n)) :
   last c cs = `c[p2].
n:nat
p1, p2:peg 3
c:configuration 3 n
cs:=ppeg p1 p2 : seq (configuration 3 n):seq (configuration 3 n)
last c cs = `c[p2]
Proof.
n:nat
p1, p2:peg 3
c:configuration 3 n
cs:=ppeg p1 p2 : seq (configuration 3 n):seq (configuration 3 n)
last c cs = `c[p2]
have HH := @rirr 3.
n:nat
p1, p2:peg 3
c:configuration 3 n
cs:=ppeg p1 p2 : seq (configuration 3 n):seq (configuration 3 n)
HH:irreflexive (rrel (n:=3))
last c cs = `c[p2]
rewrite /cs; elim: n p1 p2 c {cs} => //= [_ p2 c | n IH p1 p2 c].
HH:irreflexive (rrel (n:=3))
p2:peg 3
c:configuration 3 0
c = `c[p2]
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall p1 p2 (c : configuration 3 n),
last c (ppeg p1 p2) = `c[p2]
p1, p2:peg 3
c:configuration 3 n.+1
last c
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
`c[p2]
  by apply/ffunP=> [] [].
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall p1 p2 (c : configuration 3 n),
last c (ppeg p1 p2) = `c[p2]
p1, p2:peg 3
c:configuration 3 n.+1
last c
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
`c[p2]
by rewrite last_cat /= last_map IH perfect_liftr.
Qed.

Lemma path_ppeg n p1 p2 (cs := ppeg p1 p2 : seq (configuration 3 n)) :
  p1 != p2 -> path rmove `c[p1] cs.
n:nat
p1, p2:peg 3
cs:=ppeg p1 p2 : seq (configuration 3 n):seq (configuration 3 n)
p1 != p2 -> path rmove `c[p1] cs
Proof.
n:nat
p1, p2:peg 3
cs:=ppeg p1 p2 : seq (configuration 3 n):seq (configuration 3 n)
p1 != p2 -> path rmove `c[p1] cs
have HH := @rirr 3.
n:nat
p1, p2:peg 3
cs:=ppeg p1 p2 : seq (configuration 3 n):seq (configuration 3 n)
HH:irreflexive (rrel (n:=3))
p1 != p2 -> path rmove `c[p1] cs
rewrite /cs; elim: n p1 p2 {cs} => //= n IH p1 p2 p1Dp2.
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall p1 p2,
p1 != p2 -> path rmove `c[p1] (ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
path rmove `c[p1]
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2])
set p3 := `p[_,_].
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall p1 p2,
p1 != p2 -> path rmove `c[p1] (ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
p3:=`p[p1, p2]:peg 3
path rmove `c[p1]
  ([seq ↑[i]_p1 | i <- ppeg p1 p3] ++
   ↑[`c[p3]]_p2 :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
have p1Dp3 : p1 != p3 by rewrite eq_sym opegDl.
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall p1 p2,
p1 != p2 -> path rmove `c[p1] (ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
p3:=`p[p1, p2]:peg 3
p1Dp3:p1 != p3
path rmove `c[p1]
  ([seq ↑[i]_p1 | i <- ppeg p1 p3] ++
   ↑[`c[p3]]_p2 :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
have p3Dp2 : p3 != p2 by rewrite opegDr.
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall p1 p2,
p1 != p2 -> path rmove `c[p1] (ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
p3:=`p[p1, p2]:peg 3
p1Dp3:p1 != p3
p3Dp2:p3 != p2
path rmove `c[p1]
  ([seq ↑[i]_p1 | i <- ppeg p1 p3] ++
   ↑[`c[p3]]_p2 :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
rewrite cat_path /= -{1}[`c[p1]]cunliftrK ffunE !path_liftr // perfect_unliftr.
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall p1 p2,
p1 != p2 -> path rmove `c[p1] (ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
p3:=`p[p1, p2]:peg 3
p1Dp3:p1 != p3
p3Dp2:p3 != p2
[&& path (move (rrel (n:=3))) `c[p1] (ppeg p1 p3),
    last `c[p1] [seq ↑[i]_p1 | i <- ppeg p1 p3] `-->_r ↑[`c[p3]]_p2
  & path (move (rrel (n:=3))) `c[p3] (ppeg p3 p2)]
rewrite !IH // ?andbT /=.
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall p1 p2,
p1 != p2 -> path rmove `c[p1] (ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
p3:=`p[p1, p2]:peg 3
p1Dp3:p1 != p3
p3Dp2:p3 != p2
last `c[p1] [seq ↑[i]_p1 | i <- ppeg p1 p3] `-->_r ↑[`c[p3]]_p2
rewrite -{1}[`c[_]]cunliftrK ffunE last_map perfect_unliftr last_ppeg.
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall p1 p2,
p1 != p2 -> path rmove `c[p1] (ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
p3:=`p[p1, p2]:peg 3
p1Dp3:p1 != p3
p3Dp2:p3 != p2
↑[`c[p3]]_p1 `-->_r ↑[`c[p3]]_p2
by apply: move_liftr_perfect; rewrite // eq_sym opegDr.
Qed.

(* We can go from any perfect configuration to a perfect configuration *)
Lemma move_connect_ppeg n p1 p2 : `c[p1, n] `-->*_r `c[p2].
n:nat
p1, p2:peg 3
`c[p1 , n] `-->*_r `c[p2]
Proof.
n:nat
p1, p2:peg 3
`c[p1 , n] `-->*_r `c[p2]
case: (p1 =P p2) => [->|/eqP p1Dp2] //.
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
`c[p1] `-->*_r `c[p2]
apply/connectP; exists (ppeg p1 p2); first by apply: path_ppeg.
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
`c[p2] = last `c[p1] (ppeg p1 p2)
by rewrite last_ppeg.
Qed.

(* The proof is done by inspecting the moves that the last disk is doing in cs*)
(* We use a double  induction :                                               *)
(* The first induction is used when the path has duplicates                   *)
(* The second induction is on n and to bound recursive call                   *)
Lemma ppeg_min n p1 p2 (cs : seq (configuration 3 n)) :
  p1 != p2 -> path rmove `c[p1] cs -> last `c[p1] cs = `c[p2] ->
  (2^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2).
n:nat
p1, p2:peg 3
cs:seq (configuration 3 n)
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
Proof.
n:nat
p1, p2:peg 3
cs:seq (configuration 3 n)
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
have irrH := @rirr 3; have symH := @rsym 3.
n:nat
p1, p2:peg 3
cs:seq (configuration 3 n)
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
(* The first induction is used when the path has duplicates (1 case) *)
have [m sLm] := ubnP (size cs); elim: m => // m IHm in n p1 p2 cs sLm *.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
p1, p2:peg 3
cs:seq (configuration 3 n)
sLm:size cs < m.+1
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
elim: n p1 p2 cs sLm => /= [p1 p2 [|] //|]n IH p1 p2 cs sLm p1Dp2.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
have /= := size_ppeg n.+1 p1 p2.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
(* Is there a move of the last first disk *)
case:  path3SP => //.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
let c' := ↓[`c[p1]] in
let cs' := [seq ↓[i] | i <- cs] in
let p := `c[p1] ldisk in
cs = [seq ↑[i]_p | i <- cs'] ->
path (move (rrel (n:=3))) c' cs' ->
size
  ([seq ↑[i0]_p1 | i0 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i0]_p2 | i0 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i0]_p1
                         | i0 <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i0]_p2
                            | i0 <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
forall (cs1 : seq (configuration 3 n))
  (cs2 : seq (configuration 3 n.+1)),
let p3 := `c[p1] ldisk in
forall p4,
let p5 := `p[p3, p4] in
let c1 := ↓[`c[p1]] in
let c2 := ↑[`c[p5]]_p4 in
p3 != p4 ->
rrel p3 p4 ->
last c1 cs1 = `c[p5] ->
cs = [seq ↑[i1]_p3 | i1 <- cs1] ++ c2 :: cs2 ->
path (move (rrel (n:=3))) c1 cs1 ->
path (move (rrel (n:=3))) c2 cs2 ->
size
  ([seq ↑[i3]_p1 | i3 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i3]_p2 | i3 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i3]_p1
                         | i3 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i3]_p2
                        | i3 <- 
                      ppeg `p[p1, p2] p2])
  (* No move : impossible since p1 != p2 *)
  case: cs sLm => /= [_ _ _ _  _ /ffunP /(_ ldisk)/eqP|a cs sLm].
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
`c[p1] ldisk == `c[p2] ldisk ->
(2 ^ n.+1).-1 <= 0
              ?= iff ([::] ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
a:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
sLm:(size cs).+1 < m.+1
a :: cs =
↑[↓[a]]_(`c[p1] ldisk)
:: [seq ↑[i]_(`c[p1] ldisk)
      | i <- [seq ↓[i] | i <- cs]] ->
move (rrel (n:=3)) ↓[`c[p1]] ↓[a] &&
path (move (rrel (n:=3))) ↓[a] [seq ↓[i] | i <- cs] ->
size
  ([seq ↑[i0]_p1 | i0 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i0]_p2 | i0 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last a cs = `c[p2] ->
(2 ^ n.+1).-1 <= (size cs).+1
              ?= iff (a :: cs ==
                      [seq ↑[i0]_p1
                         | i0 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i0]_p2
                        | i0 <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
forall (cs1 : seq (configuration 3 n))
  (cs2 : seq (configuration 3 n.+1)),
let p3 := `c[p1] ldisk in
forall p4,
let p5 := `p[p3, p4] in
let c1 := ↓[`c[p1]] in
let c2 := ↑[`c[p5]]_p4 in
p3 != p4 ->
rrel p3 p4 ->
last c1 cs1 = `c[p5] ->
cs = [seq ↑[i1]_p3 | i1 <- cs1] ++ c2 :: cs2 ->
path (move (rrel (n:=3))) c1 cs1 ->
path (move (rrel (n:=3))) c2 cs2 ->
size
  ([seq ↑[i3]_p1 | i3 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i3]_p2 | i3 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i3]_p1
                         | i3 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i3]_p2
                        | i3 <- 
                      ppeg `p[p1, p2] p2])
    by rewrite !ffunE (negPf p1Dp2).
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
a:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
sLm:(size cs).+1 < m.+1
a :: cs =
↑[↓[a]]_(`c[p1] ldisk)
:: [seq ↑[i]_(`c[p1] ldisk)
      | i <- [seq ↓[i] | i <- cs]] ->
move (rrel (n:=3)) ↓[`c[p1]] ↓[a] &&
path (move (rrel (n:=3))) ↓[a] [seq ↓[i] | i <- cs] ->
size
  ([seq ↑[i0]_p1 | i0 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i0]_p2 | i0 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last a cs = `c[p2] ->
(2 ^ n.+1).-1 <= (size cs).+1
              ?= iff (a :: cs ==
                      [seq ↑[i0]_p1
                         | i0 <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i0]_p2
                            | i0 <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
forall (cs1 : seq (configuration 3 n))
  (cs2 : seq (configuration 3 n.+1)),
let p3 := `c[p1] ldisk in
forall p4,
let p5 := `p[p3, p4] in
let c1 := ↓[`c[p1]] in
let c2 := ↑[`c[p5]]_p4 in
p3 != p4 ->
rrel p3 p4 ->
last c1 cs1 = `c[p5] ->
cs = [seq ↑[i1]_p3 | i1 <- cs1] ++ c2 :: cs2 ->
path (move (rrel (n:=3))) c1 cs1 ->
path (move (rrel (n:=3))) c2 cs2 ->
size
  ([seq ↑[i3]_p1 | i3 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i3]_p2 | i3 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i3]_p1
                         | i3 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i3]_p2
                        | i3 <- 
                      ppeg `p[p1, p2] p2])
  rewrite !ffunE => acsE _ _ _ lacsE.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
a:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
sLm:(size cs).+1 < m.+1
acsE:a :: cs =
↑[↓[a]]_p1
:: [seq ↑[i]_p1 | i <- [seq ↓[i] | i <- cs]]
lacsE:last a cs = `c[p2]
(2 ^ n.+1).-1 <= (size cs).+1
              ?= iff (a :: cs ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
forall (cs1 : seq (configuration 3 n))
  (cs2 : seq (configuration 3 n.+1)),
let p3 := `c[p1] ldisk in
forall p4,
let p5 := `p[p3, p4] in
let c1 := ↓[`c[p1]] in
let c2 := ↑[`c[p5]]_p4 in
p3 != p4 ->
rrel p3 p4 ->
last c1 cs1 = `c[p5] ->
cs = [seq ↑[i1]_p3 | i1 <- cs1] ++ c2 :: cs2 ->
path (move (rrel (n:=3))) c1 cs1 ->
path (move (rrel (n:=3))) c2 cs2 ->
size
  ([seq ↑[i3]_p1 | i3 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i3]_p2 | i3 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i3]_p1
                         | i3 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i3]_p2
                        | i3 <- 
                      ppeg `p[p1, p2] p2])
  have := mem_last a cs.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
a:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
sLm:(size cs).+1 < m.+1
acsE:a :: cs =
↑[↓[a]]_p1
:: [seq ↑[i]_p1 | i <- [seq ↓[i] | i <- cs]]
lacsE:last a cs = `c[p2]
last a cs \in a :: cs ->
(2 ^ n.+1).-1 <= (size cs).+1
              ?= iff (a :: cs ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
forall (cs1 : seq (configuration 3 n))
  (cs2 : seq (configuration 3 n.+1)),
let p3 := `c[p1] ldisk in
forall p4,
let p5 := `p[p3, p4] in
let c1 := ↓[`c[p1]] in
let c2 := ↑[`c[p5]]_p4 in
p3 != p4 ->
rrel p3 p4 ->
last c1 cs1 = `c[p5] ->
cs = [seq ↑[i1]_p3 | i1 <- cs1] ++ c2 :: cs2 ->
path (move (rrel (n:=3))) c1 cs1 ->
path (move (rrel (n:=3))) c2 cs2 ->
size
  ([seq ↑[i3]_p1 | i3 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i3]_p2 | i3 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i3]_p1
                         | i3 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i3]_p2
                        | i3 <- 
                      ppeg `p[p1, p2] p2])
  rewrite acsE lacsE inE => /orP[/eqP/ffunP/(_ ldisk)/eqP|].
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
a:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
sLm:(size cs).+1 < m.+1
acsE:a :: cs =
↑[↓[a]]_p1
:: [seq ↑[i]_p1 | i <- [seq ↓[i] | i <- cs]]
lacsE:last a cs = `c[p2]
`c[p2] ldisk == ↑[↓[a]]_p1 ldisk ->
(2 ^ n.+1).-1 <= (size cs).+1
              ?= iff (↑[↓[a]]_p1
                      :: [seq ↑[i]_p1
                            | i <- [seq ↓[i]
                                      | i <- cs]] ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
a:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
sLm:(size cs).+1 < m.+1
acsE:a :: cs =
↑[↓[a]]_p1
:: [seq ↑[i]_p1 | i <- [seq ↓[i] | i <- cs]]
lacsE:last a cs = `c[p2]
`c[p2] \in [seq ↑[i]_p1 | i <- [seq ↓[i] | i <- cs]] ->
(2 ^ n.+1).-1 <= (size cs).+1
              ?= iff (↑[↓[a]]_p1
                      :: [
                      seq ↑[i]_p1
                        | i <- [
                      seq ↓[i] | i <- cs]] ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
forall (cs1 : seq (configuration 3 n))
  (cs2 : seq (configuration 3 n.+1)),
let p3 := `c[p1] ldisk in
forall p4,
let p5 := `p[p3, p4] in
let c1 := ↓[`c[p1]] in
let c2 := ↑[`c[p5]]_p4 in
p3 != p4 ->
rrel p3 p4 ->
last c1 cs1 = `c[p5] ->
cs = [seq ↑[i1]_p3 | i1 <- cs1] ++ c2 :: cs2 ->
path (move (rrel (n:=3))) c1 cs1 ->
path (move (rrel (n:=3))) c2 cs2 ->
size
  ([seq ↑[i3]_p1 | i3 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i3]_p2 | i3 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i3]_p1
                         | i3 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i3]_p2
                        | i3 <- 
                      ppeg `p[p1, p2] p2])
    by rewrite cliftr_ldisk !ffunE eq_sym (negPf p1Dp2).
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
a:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
sLm:(size cs).+1 < m.+1
acsE:a :: cs =
↑[↓[a]]_p1
:: [seq ↑[i]_p1 | i <- [seq ↓[i] | i <- cs]]
lacsE:last a cs = `c[p2]
`c[p2] \in [seq ↑[i]_p1 | i <- [seq ↓[i] | i <- cs]] ->
(2 ^ n.+1).-1 <= (size cs).+1
              ?= iff (↑[↓[a]]_p1
                      :: [seq ↑[i]_p1
                            | i <- [seq ↓[i]
                                      | i <- cs]] ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
forall (cs1 : seq (configuration 3 n))
  (cs2 : seq (configuration 3 n.+1)),
let p3 := `c[p1] ldisk in
forall p4,
let p5 := `p[p3, p4] in
let c1 := ↓[`c[p1]] in
let c2 := ↑[`c[p5]]_p4 in
p3 != p4 ->
rrel p3 p4 ->
last c1 cs1 = `c[p5] ->
cs = [seq ↑[i1]_p3 | i1 <- cs1] ++ c2 :: cs2 ->
path (move (rrel (n:=3))) c1 cs1 ->
path (move (rrel (n:=3))) c2 cs2 ->
size
  ([seq ↑[i3]_p1 | i3 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i3]_p2 | i3 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i3]_p1
                         | i3 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i3]_p2
                        | i3 <- 
                      ppeg `p[p1, p2] p2])
  case/mapP => c _ /ffunP /(_ ldisk)/eqP.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
p1Dp2:p1 != p2
a:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
sLm:(size cs).+1 < m.+1
acsE:a :: cs =
↑[↓[a]]_p1
:: [seq ↑[i]_p1 | i <- [seq ↓[i] | i <- cs]]
lacsE:last a cs = `c[p2]
c:finfun_eqType (ordinal_finType n)
  (ordinal_eqType 3)
`c[p2] ldisk == ↑[c]_p1 ldisk ->
(2 ^ n.+1).-1 <= (size cs).+1
              ?= iff (↑[↓[a]]_p1
                      :: [seq ↑[i]_p1
                            | i <- [seq ↓[i]
                                      | i <- cs]] ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
forall (cs1 : seq (configuration 3 n))
  (cs2 : seq (configuration 3 n.+1)),
let p3 := `c[p1] ldisk in
forall p4,
let p5 := `p[p3, p4] in
let c1 := ↓[`c[p1]] in
let c2 := ↑[`c[p5]]_p4 in
p3 != p4 ->
rrel p3 p4 ->
last c1 cs1 = `c[p5] ->
cs = [seq ↑[i1]_p3 | i1 <- cs1] ++ c2 :: cs2 ->
path (move (rrel (n:=3))) c1 cs1 ->
path (move (rrel (n:=3))) c2 cs2 ->
size
  ([seq ↑[i3]_p1 | i3 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i3]_p2 | i3 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i3]_p1
                         | i3 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i3]_p2
                        | i3 <- 
                      ppeg `p[p1, p2] p2])
  by rewrite cliftr_ldisk ffunE eq_sym (negPf p1Dp2).
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
forall (cs1 : seq (configuration 3 n))
  (cs2 : seq (configuration 3 n.+1)),
let p3 := `c[p1] ldisk in
forall p4,
let p5 := `p[p3, p4] in
let c1 := ↓[`c[p1]] in
let c2 := ↑[`c[p5]]_p4 in
p3 != p4 ->
rrel p3 p4 ->
last c1 cs1 = `c[p5] ->
cs = [seq ↑[i1]_p3 | i1 <- cs1] ++ c2 :: cs2 ->
path (move (rrel (n:=3))) c1 cs1 ->
path (move (rrel (n:=3))) c2 cs2 ->
size
  ([seq ↑[i3]_p1 | i3 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i3]_p2 | i3 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i3]_p1
                         | i3 <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i3]_p2
                            | i3 <- ppeg `p[p1, p2] p2])
(* There is a move from p1 to p3 *)
rewrite !ffunE perfect_unliftr /= => cs1 cs2 p3 p1Dp3 _.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
last `c[p1] cs1 = `c[`p[p1, p3]] ->
cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2 ->
path (move (rrel (n:=3))) `c[p1] cs1 ->
path (move (rrel (n:=3))) ↑[`c[`p[p1, p3]]]_p3 cs2 ->
size
  ([seq ↑[i1]_p1 | i1 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i1]_p2 | i1 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i1]_p1
                         | i1 <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i1]_p2
                            | i1 <- ppeg `p[p1, p2] p2])
move=> p1cs1Lp1p3 csE p1Pcs1.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
path (move (rrel (n:=3))) ↑[`c[`p[p1, p3]]]_p3 cs2 ->
size
  ([seq ↑[i0]_p1 | i0 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i0]_p2 | i0 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i0]_p1
                         | i0 <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i0]_p2
                            | i0 <- ppeg `p[p1, p2] p2])
have Scs1 : size cs1 < m.+1.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
size cs1 < m.+1
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
path (move (rrel (n:=3))) ↑[`c[`p[p1, p3]]]_p3 cs2 ->
size
  ([seq ↑[i0]_p1 | i0 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i0]_p2 | i0 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i0]_p1
                         | i0 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i0]_p2
                        | i0 <- 
                      ppeg `p[p1, p2] p2])
  by apply: leq_trans sLm; rewrite csE size_cat size_map ltnS leq_addr.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
path (move (rrel (n:=3))) ↑[`c[`p[p1, p3]]]_p3 cs2 ->
size
  ([seq ↑[i0]_p1 | i0 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i0]_p2 | i0 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i0]_p1
                         | i0 <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i0]_p2
                            | i0 <- ppeg `p[p1, p2] p2])
have p1Dp1p3 : p1 != `p[p1, p3] by rewrite eq_sym opegDl.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
path (move (rrel (n:=3))) ↑[`c[`p[p1, p3]]]_p3 cs2 ->
size
  ([seq ↑[i0]_p1 | i0 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i0]_p2 | i0 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i0]_p1
                         | i0 <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i0]_p2
                            | i0 <- ppeg `p[p1, p2] p2])
(* After the first move, last disk is on p3, the other disk is `p[p1, p3] *)
have HL1 := IH _ _ _ Scs1 p1Dp1p3 p1Pcs1 p1cs1Lp1p3.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
path (move (rrel (n:=3))) ↑[`c[`p[p1, p3]]]_p3 cs2 ->
size
  ([seq ↑[i0]_p1 | i0 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i0]_p2 | i0 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i0]_p1
                         | i0 <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i0]_p2
                            | i0 <- ppeg `p[p1, p2] p2])
have n2E : (2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
path (move (rrel (n:=3))) ↑[`c[`p[p1, p3]]]_p3 cs2 ->
size
  ([seq ↑[i0]_p1 | i0 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i0]_p2 | i0 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i0]_p1
                         | i0 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i0]_p2
                        | i0 <- 
                      ppeg `p[p1, p2] p2])
  by rewrite expnS mul2n -addnn -[2 ^ n]prednK ?expn_gt0.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
path (move (rrel (n:=3))) ↑[`c[`p[p1, p3]]]_p3 cs2 ->
size
  ([seq ↑[i0]_p1 | i0 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i0]_p2 | i0 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i0]_p1
                         | i0 <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i0]_p2
                            | i0 <- ppeg `p[p1, p2] p2])
(* Is there another move *)
case:  path3SP => //=.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs2 =
[seq ↑[i]_(↑[`c[`p[p1, p3]]]_p3 ldisk)
   | i <- [seq ↓[i] | i <- cs2]] ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3]
  [seq ↓[i] | i <- cs2] ->
true ->
size
  ([seq ↑[i1]_p1 | i1 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i1]_p2 | i1 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i1]_p1
                         | i1 <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i1]_p2
                            | i1 <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
  (* there is no move so p3 = p2 and simple induction should make it *)
  rewrite cliftr_ldisk cliftrK /=.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs2 = [seq ↑[i]_p3 | i <- [seq ↓[i] | i <- cs2]] ->
path (move (rrel (n:=3))) `c[`p[p1, p3]]
  [seq ↓[i] | i <- cs2] ->
true ->
size
  ([seq ↑[i1]_p1 | i1 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i1]_p2 | i1 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i1]_p1
                         | i1 <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i1]_p2
                            | i1 <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
  move=> cs2E p1p3Pcs2 _ sH _ p1csLp2.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs2E:cs2 =
[seq ↑[i]_p3 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p3]]
  [seq ↓[i] | i <- cs2]
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
  have p3E : p3 = p2.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs2E:cs2 =
[seq ↑[i]_p3 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p3]]
  [seq ↓[i] | i <- cs2]
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3 = p2
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs2E:cs2 =
[seq ↑[i]_p3 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p3]]
  [seq ↓[i] | i <- cs2]
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
    move: p1csLp2; rewrite csE cs2E last_cat /= last_map.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs2E:cs2 =
[seq ↑[i]_p3 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p3]]
  [seq ↓[i] | i <- cs2]
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
↑[last `c[`p[p1, p3]] [seq ↓[i] | i <- cs2]]_p3 =
`c[p2] -> p3 = p2
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs2E:cs2 =
[seq ↑[i]_p3 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p3]]
  [seq ↓[i] | i <- cs2]
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
    by move=> /ffunP /(_ ldisk); rewrite cliftr_ldisk ffunE.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs2E:cs2 =
[seq ↑[i]_p3 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p3]]
  [seq ↓[i] | i <- cs2]
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
  rewrite p3E in p1cs1Lp1p3 csE HL1 cs2E p1p3Pcs2.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p2])
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
  have Scs2 : size [seq ↓[i] | i <- cs2] < m.+1.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p2])
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
size [seq ↓[i] | i <- cs2] < m.+1
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p2])
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
    apply: leq_trans sLm.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p2])
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
size [seq ↓[i] | i <- cs2] < (size cs).+1
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p2])
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
    by rewrite csE size_cat !size_map /= ltnS -addSnnS leq_addl.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p2])
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
  have p1p2Lcs2 : last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] = `c[p2].
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p2])
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] = `c[p2]
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p2])
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
    rewrite -[`c[_]](cliftrK p2) last_map.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p2])
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
↓[last ↑[`c[`p[p1, p2]]]_p2 cs2] = `c[p2]
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p2])
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
    by have := p1csLp2; rewrite csE last_cat /= => ->; rewrite perfect_unliftr.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p2])
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
  have HL2 := IH _ _ _ Scs2 (opegDr _ _) p1p3Pcs2 p1p2Lcs2.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p2])
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL2:(2 ^ n).-1 <= size [seq ↓[i] | i <- cs2]
           ?= iff ([seq ↓[i] | i <- cs2] ==
                   ppeg `p[p1, p2] p2)
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
  move/leqifP : HL1; case: eqP => [<- /eqP HL1|_ HL1].
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL2:(2 ^ n).-1 <= size [seq ↓[i] | i <- cs2]
           ?= iff ([seq ↓[i] | i <- cs2] ==
                   ppeg `p[p1, p2] p2)
HL1:(2 ^ n).-1 = size cs1
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1 | i <- cs1] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL2:(2 ^ n).-1 <= size [seq ↓[i] | i <- cs2]
           ?= iff ([seq ↓[i] | i <- cs2] ==
                   ppeg `p[p1, p2] p2)
HL1:(2 ^ n).-1 < size cs1
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
    move/leqifP : HL2; case: eqP => [<- /eqP HL2|_ HL2].
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL1:(2 ^ n).-1 = size cs1
HL2:(2 ^ n).-1 = size [seq ↓[i] | i <- cs2]
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1 | i <- cs1] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- [seq ↓[i]
                                      | i <- cs2]])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL1:(2 ^ n).-1 = size cs1
HL2:(2 ^ n).-1 < size [seq ↓[i] | i <- cs2]
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1 | i <- cs1] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL2:(2 ^ n).-1 <= size [seq ↓[i] | i <- cs2]
           ?= iff ([seq ↓[i] | i <- cs2] ==
                   ppeg `p[p1, p2] p2)
HL1:(2 ^ n).-1 < size cs1
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
      rewrite size_map in HL2.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL1:(2 ^ n).-1 = size cs1
HL2:(2 ^ n).-1 = size cs2
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1 | i <- cs1] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- [seq ↓[i]
                                      | i <- cs2]])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL1:(2 ^ n).-1 = size cs1
HL2:(2 ^ n).-1 < size [seq ↓[i] | i <- cs2]
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1 | i <- cs1] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL2:(2 ^ n).-1 <= size [seq ↓[i] | i <- cs2]
           ?= iff ([seq ↓[i] | i <- cs2] ==
                   ppeg `p[p1, p2] p2)
HL1:(2 ^ n).-1 < size cs1
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
      rewrite csE size_cat /= size_map -HL1 -HL2 -cs2E eqxx //= n2E.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL1:(2 ^ n).-1 = size cs1
HL2:(2 ^ n).-1 = size cs2
(2 ^ n).-1 + (2 ^ n).-1.+1 <= (2 ^ n).-1 +
                              (2 ^ n).-1.+1
                           ?= iff true
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL1:(2 ^ n).-1 = size cs1
HL2:(2 ^ n).-1 < size [seq ↓[i] | i <- cs2]
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1 | i <- cs1] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL2:(2 ^ n).-1 <= size [seq ↓[i] | i <- cs2]
           ?= iff ([seq ↓[i] | i <- cs2] ==
                   ppeg `p[p1, p2] p2)
HL1:(2 ^ n).-1 < size cs1
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
      by apply/leqifP.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL1:(2 ^ n).-1 = size cs1
HL2:(2 ^ n).-1 < size [seq ↓[i] | i <- cs2]
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1 | i <- cs1] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL2:(2 ^ n).-1 <= size [seq ↓[i] | i <- cs2]
           ?= iff ([seq ↓[i] | i <- cs2] ==
                   ppeg `p[p1, p2] p2)
HL1:(2 ^ n).-1 < size cs1
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
    apply/leqifP; case: eqP => [/(congr1 size)->|_].
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL1:(2 ^ n).-1 = size cs1
HL2:(2 ^ n).-1 < size [seq ↓[i] | i <- cs2]
(2 ^ n.+1).-1 ==
size
  ([seq ↑[i]_p1 | i <- cs1] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL1:(2 ^ n).-1 = size cs1
HL2:(2 ^ n).-1 < size [seq ↓[i] | i <- cs2]
(2 ^ n.+1).-1 < size cs
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL2:(2 ^ n).-1 <= size [seq ↓[i] | i <- cs2]
           ?= iff ([seq ↓[i] | i <- cs2] ==
                   ppeg `p[p1, p2] p2)
HL1:(2 ^ n).-1 < size cs1
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
      by rewrite -sH !size_cat /= !size_map -HL1 size_ppeg.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL1:(2 ^ n).-1 = size cs1
HL2:(2 ^ n).-1 < size [seq ↓[i] | i <- cs2]
(2 ^ n.+1).-1 < size cs
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL2:(2 ^ n).-1 <= size [seq ↓[i] | i <- cs2]
           ?= iff ([seq ↓[i] | i <- cs2] ==
                   ppeg `p[p1, p2] p2)
HL1:(2 ^ n).-1 < size cs1
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
    rewrite csE size_cat /= !size_map in HL2 *.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL1:(2 ^ n).-1 = size cs1
HL2:(2 ^ n).-1 < size cs2
(2 ^ n.+1).-1 < size cs1 + (size cs2).+1
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL2:(2 ^ n).-1 <= size [seq ↓[i] | i <- cs2]
           ?= iff ([seq ↓[i] | i <- cs2] ==
                   ppeg `p[p1, p2] p2)
HL1:(2 ^ n).-1 < size cs1
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i]_p2
                        | i <- 
                      ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
    by rewrite -HL1 n2E ltn_add2l.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL2:(2 ^ n).-1 <= size [seq ↓[i] | i <- cs2]
           ?= iff ([seq ↓[i] | i <- cs2] ==
                   ppeg `p[p1, p2] p2)
HL1:(2 ^ n).-1 < size cs1
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
  apply/leqifP; case: eqP => [/(congr1 size)->|_]; first by rewrite -sH.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL2:(2 ^ n).-1 <= size [seq ↓[i] | i <- cs2]
           ?= iff ([seq ↓[i] | i <- cs2] ==
                   ppeg `p[p1, p2] p2)
HL1:(2 ^ n).-1 < size cs1
(2 ^ n.+1).-1 < size cs
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
  rewrite csE size_cat /= !size_map in HL2 *.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p3E:p3 = p2
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p2]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p2]]]_p2 :: cs2
cs2E:cs2 =
[seq ↑[i]_p2 | i <- [seq ↓[i] | i <- cs2]]
p1p3Pcs2:path (move (rrel (n:=3))) `c[`p[p1, p2]]
  [seq ↓[i] | i <- cs2]
Scs2:size [seq ↓[i] | i <- cs2] < m.+1
p1p2Lcs2:last `c[`p[p1, p2]] [seq ↓[i] | i <- cs2] =
`c[p2]
HL1:(2 ^ n).-1 < size cs1
HL2:(2 ^ n).-1 <= size cs2
           ?= iff ([seq ↓[i] | i <- cs2] ==
                   ppeg `p[p1, p2] p2)
(2 ^ n.+1).-1 < size cs1 + (size cs2).+1
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) 
  p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- 
                       ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [
                      seq ↑[i4]_p2
                        | i4 <- 
                      ppeg `p[p1, p2] p2])
  by rewrite n2E -addSn leq_add // ltnS HL2.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
forall (cs1 : seq (configuration 3 n))
  (cs3 : seq (configuration 3 n.+1)) p4,
↑[`c[`p[p1, p3]]]_p3 ldisk != p4 ->
rrel (↑[`c[`p[p1, p3]]]_p3 ldisk) p4 ->
last ↓[↑[`c[`p[p1, p3]]]_p3] cs1 =
`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]] ->
cs2 =
[seq ↑[i1]_(↑[`c[`p[p1, p3]]]_p3 ldisk) | i1 <- cs1] ++
↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 :: cs3 ->
path (move (rrel (n:=3))) ↓[↑[`c[`p[p1, p3]]]_p3] cs1 ->
path (move (rrel (n:=3)))
  ↑[`c[`p[↑[`c[`p[p1, p3]]]_p3 ldisk, p4]]]_p4 cs3 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i4]_p2
                            | i4 <- ppeg `p[p1, p2] p2])
move=> cs3 cs4 p4; rewrite -!/rmove !cliftr_ldisk cliftrK.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3 != p4 ->
rrel p3 p4 ->
last `c[`p[p1, p3]] cs3 = `c[`p[p3, p4]] ->
cs2 =
[seq ↑[i1]_p3 | i1 <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4 ->
path rmove `c[`p[p1, p3]] cs3 ->
path rmove ↑[`c[`p[p3, p4]]]_p4 cs4 ->
true ->
size
  ([seq ↑[i4]_p1 | i4 <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i4]_p2 | i4 <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1 ->
true ->
last `c[p1] cs = `c[p2] ->
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i4]_p1
                         | i4 <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i4]_p2
                            | i4 <- ppeg `p[p1, p2] p2])
move => p3Dp4 _ p1p3cs3Lp3p4 cs2E p1p3Pcs3 p3p4Pcs4 _ sH _ p1csLp2.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
(2 ^ n.+1).-1 <= size cs
              ?= iff (cs ==
                      [seq ↑[i]_p1
                         | i <- ppeg p1 `p[p1, p2]] ++
                      ↑[`c[`p[p1, p2]]]_p2
                      :: [seq ↑[i]_p2
                            | i <- ppeg `p[p1, p2] p2])
(* we did two moves of the largest disk so we cannot be = *)
apply/leqifP; case: eqP => [/(congr1 size)|_].
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
size cs =
size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) ->
(2 ^ n.+1).-1 == size cs
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
(2 ^ n.+1).-1 < size cs
  by rewrite size_cat /= !size_map !size_ppeg n2E => ->.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
(2 ^ n.+1).-1 < size cs
(* Did we come back to p1 *)
case: (p4 =P p1) => [p4Ep1 | /eqP p4Dp1].
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
(2 ^ n.+1).-1 < size cs
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n.+1).-1 < size cs
  rewrite p4Ep1 in p3Dp4 p1p3cs3Lp3p4 cs2E p3p4Pcs4.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
(2 ^ n.+1).-1 < size cs
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n.+1).-1 < size cs
  (* if so cs has a repetition so we can use IHm *)
  pose cs' := [seq ↑[i]_p1 | i <- cs1] ++ cs4.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
(2 ^ n.+1).-1 < size cs
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n.+1).-1 < size cs
  have Scs' : size cs' < size cs.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
size cs' < size cs
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
Scs':size cs' < size cs
(2 ^ n.+1).-1 < size cs
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n.+1).-1 < size cs
    rewrite csE /cs' cs2E !size_cat /= size_cat !size_map /=.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
size cs1 + size cs4 <
size cs1 + (size cs3 + (size cs4).+1).+1
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
Scs':size cs' < size cs
(2 ^ n.+1).-1 < size cs
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n.+1).-1 < size cs
    by rewrite ltn_add2l -!addSnnS ltnS leq_addl.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
Scs':size cs' < size cs
(2 ^ n.+1).-1 < size cs
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n.+1).-1 < size cs
  apply: leq_trans (Scs'); rewrite ltnS.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
Scs':size cs' < size cs
(2 ^ n.+1).-1 <= size cs'
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n.+1).-1 < size cs
  rewrite (IHm _ p1 p2) //; first by rewrite (leq_trans Scs').
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
Scs':size cs' < size cs
path rmove `c[p1] cs'
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
Scs':size cs' < size cs
last `c[p1] cs' = `c[p2]
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n.+1).-1 < size cs
    rewrite cat_path /= -[`c[_]](cunliftrK) ffunE perfect_unliftr.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
Scs':size cs' < size cs
path rmove ↑[`c[p1]]_p1 [seq ↑[i]_p1 | i <- cs1] &&
path rmove
  (last ↑[`c[p1]]_p1 [seq ↑[i]_p1 | i <- cs1]) cs4
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
Scs':size cs' < size cs
last `c[p1] cs' = `c[p2]
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n.+1).-1 < size cs
    by rewrite path_liftr // p1Pcs1 /= last_map p1cs1Lp1p3 opeg_sym.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
Scs':size cs' < size cs
last `c[p1] cs' = `c[p2]
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n.+1).-1 < size cs
  rewrite last_cat /= -[`c[_]](cunliftrK) ffunE perfect_unliftr.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
Scs':size cs' < size cs
last (last ↑[`c[p1]]_p1 [seq ↑[i]_p1 | i <- cs1]) cs4 =
`c[p2]
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n.+1).-1 < size cs
  rewrite last_map p1cs1Lp1p3.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
Scs':size cs' < size cs
last ↑[`c[`p[p1, p3]]]_p1 cs4 = `c[p2]
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n.+1).-1 < size cs
  have := p1csLp2.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Ep1:p4 = p1
p3Dp4:p3 != p1
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p1]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p1]]]_p1 :: cs4
p3p4Pcs4:path rmove ↑[`c[`p[p3, p1]]]_p1 cs4
cs':=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
Scs':size cs' < size cs
last `c[p1] cs = `c[p2] ->
last ↑[`c[`p[p1, p3]]]_p1 cs4 = `c[p2]
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n.+1).-1 < size cs
  by rewrite csE cs2E !last_cat /= last_cat /= opeg_sym.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n.+1).-1 < size cs
rewrite csE cs2E size_cat /= size_cat /= !size_map n2E !addnS !ltnS addnA.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n).-1 + (2 ^ n).-1 <=
size cs1 + size cs3 + size cs4
apply: leq_trans (leq_add _ _) (leq_addr _ _); first by rewrite HL1.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
(2 ^ n).-1 <= size cs3
have Scs3 : size cs3 < m.+1.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
size cs3 < m.+1
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
Scs3:size cs3 < m.+1
(2 ^ n).-1 <= size cs3
  apply: leq_trans sLm.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
size cs3 < (size cs).+1
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
Scs3:size cs3 < m.+1
(2 ^ n).-1 <= size cs3
  rewrite csE cs2E size_cat /= size_cat /= !size_map addnC ltnS addnS !addSnnS.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
size cs3 <= size cs3 + size cs4 + (size cs1).+2
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
Scs3:size cs3 < m.+1
(2 ^ n).-1 <= size cs3
  by rewrite -addnA leq_addr.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
Scs3:size cs3 < m.+1
(2 ^ n).-1 <= size cs3
rewrite (IH _ _ _ Scs3 _ p1p3Pcs3 p1p3cs3Lp3p4) //.
irrH:irreflexive (rrel (n:=3))
symH:symmetric (rrel (n:=3))
m:nat
IHm:forall (n : nat) p1 p2
  (cs : seq (configuration 3 n)),
size cs < m ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
n:nat
IH:forall p1 p2 (cs : seq (configuration 3 n)),
size cs < m.+1 ->
p1 != p2 ->
path rmove `c[p1] cs ->
last `c[p1] cs = `c[p2] ->
(2 ^ n).-1 <= size cs ?= iff (cs == ppeg p1 p2)
p1, p2:peg 3
cs:seq (configuration 3 n.+1)
sLm:size cs < m.+1
p1Dp2:p1 != p2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p3:peg 3
p1Dp3:p1 != p3
p1cs1Lp1p3:last `c[p1] cs1 = `c[`p[p1, p3]]
csE:cs =
[seq ↑[i]_p1 | i <- cs1] ++
↑[`c[`p[p1, p3]]]_p3 :: cs2
p1Pcs1:path (move (rrel (n:=3))) `c[p1] cs1
Scs1:size cs1 < m.+1
p1Dp1p3:p1 != `p[p1, p3]
HL1:(2 ^ n).-1 <= size cs1
           ?= iff (cs1 == ppeg p1 `p[p1, p3])
n2E:(2 ^ n.+1).-1 = (2 ^ n).-1 + (2 ^ n).-1.+1
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p3Dp4:p3 != p4
p1p3cs3Lp3p4:last `c[`p[p1, p3]] cs3 =
`c[`p[p3, p4]]
cs2E:cs2 =
[seq ↑[i]_p3 | i <- cs3] ++
↑[`c[`p[p3, p4]]]_p4 :: cs4
p1p3Pcs3:path rmove `c[`p[p1, p3]] cs3
p3p4Pcs4:path rmove ↑[`c[`p[p3, p4]]]_p4 cs4
sH:size
  ([seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
   ↑[`c[`p[p1, p2]]]_p2
   :: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]) =
(2 ^ n.+1).-1
p1csLp2:last `c[p1] cs = `c[p2]
p4Dp1:p4 != p1
Scs3:size cs3 < m.+1
`p[p1, p3] != `p[p3, p4]
by rewrite opeg3E // eq_sym opeg3E // eq_sym p1Dp3 p4Dp1 eq_sym opegDl.
Qed.

Lemma gdist_rhanoi3p n p1 p2 :
  `d[`c[p1, n], `c[p2]]_rmove = (2^ n).-1 * (p1 != p2).
n:nat
p1, p2:peg 3
`d[`c[p1 , n], `c[p2]]_rmove = (2 ^ n).-1 * (p1 != p2)
Proof.
n:nat
p1, p2:peg 3
`d[`c[p1 , n], `c[p2]]_rmove = (2 ^ n).-1 * (p1 != p2)
case: eqP => [<-|/eqP p1Dp2]; first by rewrite muln0 gdist0.
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
`d[`c[p1], `c[p2]]_rmove = (2 ^ n).-1 * ~~ false
rewrite muln1.
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
`d[`c[p1], `c[p2]]_rmove = (2 ^ n).-1
apply/eqP; rewrite eqn_leq -(size_ppeg n p1 p2).
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
`d[`c[p1], `c[p2]]_rmove <= size (ppeg p1 p2) <=
`d[`c[p1], `c[p2]]_rmove
rewrite gdist_path_le //=; last 2 first.
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
path rmove `c[p1] (ppeg p1 p2)
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
last `c[p1] (ppeg p1 p2) = `c[p2]
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
size (ppeg p1 p2) <= `d[`c[p1], `c[p2]]_rmove
-
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
path rmove `c[p1] (ppeg p1 p2)
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
last `c[p1] (ppeg p1 p2) = `c[p2]
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
size (ppeg p1 p2) <= `d[`c[p1], `c[p2]]_rmove by apply: path_ppeg.
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
last `c[p1] (ppeg p1 p2) = `c[p2]
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
size (ppeg p1 p2) <= `d[`c[p1], `c[p2]]_rmove
-
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
last `c[p1] (ppeg p1 p2) = `c[p2]
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
size (ppeg p1 p2) <= `d[`c[p1], `c[p2]]_rmove by rewrite last_ppeg.
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
size (ppeg p1 p2) <= `d[`c[p1], `c[p2]]_rmove
have /gpath_connect [cs csH] : connect rmove `c[p1, n] `c[p2].
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
`c[p1] `-->*_r `c[p2]
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
cs:seq
  (finfun_finType (ordinal_finType n)
     (ordinal_finType 3))
csH:gpath rmove `c[p1] `c[p2] cs
size (ppeg p1 p2) <= `d[`c[p1], `c[p2]]_rmove
  by apply: move_connect_ppeg.
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
cs:seq
  (finfun_finType (ordinal_finType n)
     (ordinal_finType 3))
csH:gpath rmove `c[p1] `c[p2] cs
size (ppeg p1 p2) <= `d[`c[p1], `c[p2]]_rmove
rewrite size_ppeg (gpath_dist csH) /=.
n:nat
p1, p2:peg 3
p1Dp2:p1 != p2
cs:seq
  (finfun_finType (ordinal_finType n)
     (ordinal_finType 3))
csH:gpath rmove `c[p1] `c[p2] cs
(2 ^ n).-1 <= size cs
apply: ppeg_min p1Dp2 (gpath_path csH) _ => //.
n:nat
p1, p2:peg 3
cs:seq
  (finfun_finType (ordinal_finType n)
     (ordinal_finType 3))
csH:gpath rmove `c[p1] `c[p2] cs
last `c[p1] cs = `c[p2]
by apply: gpath_last csH.
Qed.

(*****************************************************************************)
(* Function that builds a path from a configuration to a peg                 *)
(*****************************************************************************)

Fixpoint rpeg {n : nat} :=
  if n is n1.+1 return configuration 3 n -> peg 3 -> seq (configuration 3 n) 
  then
    fun c p =>
    let p1 := c ldisk in
    if p1 == p then [seq ↑[i]_p | i <- rpeg ↓[c] p] else
    let p2 := `p[p1, p] in
    [seq ↑[i]_p1 | i <- rpeg ↓[c] p2] ++
    [seq ↑[i]_p | i <- `c[p2] :: ppeg p2 p]
  else fun _ _ => [::].

Lemma rpeg_perfect n p : rpeg (`c[p]) p = [::] :> seq (configuration 3 n).
n:nat
p:peg 3
rpeg `c[p] p = [::]
Proof.
n:nat
p:peg 3
rpeg `c[p] p = [::]
elim: n => //= n IH.
p:peg 3
n:nat
IH:rpeg `c[p] p = [::]
(if `c[p] ldisk == p
 then [seq ↑[i]_p | i <- rpeg ↓[`c[p]] p]
 else
  [seq ↑[i]_(`c[p] ldisk)
     | i <- rpeg ↓[`c[p]] `p[`c[p] ldisk, p]] ++
  ↑[`c[`p[`c[p] ldisk, p]]]_p
  :: [seq ↑[i]_p | i <- ppeg `p[`c[p] ldisk, p] p]) =
[::]
by rewrite ffunE eqxx perfect_unliftr IH.
Qed.

Lemma rpeg_nil_inv n c p : rpeg c p = [::] -> c = `c[p] :> configuration _ n.
n:nat
c:configuration 3 n
p:peg 3
rpeg c p = [::] -> c = `c[p]
Proof.
n:nat
c:configuration 3 n
p:peg 3
rpeg c p = [::] -> c = `c[p]
elim: n c => [c _|n IH c] /=; first by apply/ffunP=> [] [].
p:peg 3
n:nat
IH:forall c : configuration 3 n,
rpeg c p = [::] -> c = `c[p]
c:configuration 3 n.+1
(if c ldisk == p
 then [seq ↑[i]_p | i <- rpeg ↓[c] p]
 else
  [seq ↑[i]_(c ldisk) | i <- rpeg ↓[c] `p[c ldisk, p]] ++
  ↑[`c[`p[c ldisk, p]]]_p
  :: [seq ↑[i]_p | i <- ppeg `p[c ldisk, p] p]) = [::] ->
c = `c[p]
case: eqP => [H | H] //=; last by case: map.
p:peg 3
n:nat
IH:forall c : configuration 3 n,
rpeg c p = [::] -> c = `c[p]
c:configuration 3 n.+1
H:c ldisk = p
[seq ↑[i]_p | i <- rpeg ↓[c] p] = [::] -> c = `c[p]
rewrite -{2}[c]cunliftrK.
p:peg 3
n:nat
IH:forall c : configuration 3 n,
rpeg c p = [::] -> c = `c[p]
c:configuration 3 n.+1
H:c ldisk = p
[seq ↑[i]_p | i <- rpeg ↓[c] p] = [::] ->
↑[↓[c]]_(c ldisk) = `c[p]
case: rpeg (IH (↓[c])) => // -> // _.
p:peg 3
n:nat
IH:forall c : configuration 3 n,
rpeg c p = [::] -> c = `c[p]
c:configuration 3 n.+1
H:c ldisk = p
↑[`c[p]]_(c ldisk) = `c[p]
by rewrite H perfect_liftr.
Qed.

Lemma rpeg_ppeg n p1 p2 : p1 != p2 -> rpeg `c[p1, n] p2  = ppeg p1 p2.
n:nat
p1, p2:peg 3
p1 != p2 -> rpeg `c[p1 , n] p2 = ppeg p1 p2 
Proof.
n:nat
p1, p2:peg 3
p1 != p2 -> rpeg `c[p1 , n] p2 = ppeg p1 p2
elim: n p1 p2 => //= n IH p1 p2 p1Dp2.
n:nat
IH:forall p1 p2,
p1 != p2 -> rpeg `c[p1] p2 = ppeg p1 p2
p1, p2:peg 3
p1Dp2:p1 != p2
(if `c[p1] ldisk == p2
 then [seq ↑[i]_p2 | i <- rpeg ↓[`c[p1]] p2]
 else
  [seq ↑[i]_(`c[p1] ldisk)
     | i <- rpeg ↓[`c[p1]] `p[`c[p1] ldisk, p2]] ++
  ↑[`c[`p[`c[p1] ldisk, p2]]]_p2
  :: [seq ↑[i]_p2 | i <- ppeg `p[`c[p1] ldisk, p2] p2]) =
[seq ↑[i]_p1 | i <- ppeg p1 `p[p1, p2]] ++
↑[`c[`p[p1, p2]]]_p2
:: [seq ↑[i]_p2 | i <- ppeg `p[p1, p2] p2]
by rewrite ffunE perfect_unliftr (negPf p1Dp2) !IH // eq_sym opegDl.
Qed.

Lemma last_rpeg n (c : configuration 3 n) p (cs := rpeg c p) :
  last c cs = `c[p].
n:nat
c:configuration 3 n
p:peg 3
cs:=rpeg c p:seq (configuration 3 n)
last c cs = `c[p]
Proof.
n:nat
c:configuration 3 n
p:peg 3
cs:=rpeg c p:seq (configuration 3 n)
last c cs = `c[p]
rewrite /cs; elim: n c p {cs} => /= [c p| n IH c p].
c:configuration 3 0
p:peg 3
c = `c[p]
n:nat
IH:forall (c : configuration 3 n) p,
last c (rpeg c p) = `c[p]
c:configuration 3 n.+1
p:peg 3
last c
  (if c ldisk == p
   then [seq ↑[i]_p | i <- rpeg ↓[c] p]
   else
    [seq ↑[i]_(c ldisk)
       | i <- rpeg ↓[c] `p[c ldisk, p]] ++
    ↑[`c[`p[c ldisk, p]]]_p
    :: [seq ↑[i]_p | i <- ppeg `p[c ldisk, p] p]) =
`c[p]
  by apply/ffunP=> [] [].
n:nat
IH:forall (c : configuration 3 n) p,
last c (rpeg c p) = `c[p]
c:configuration 3 n.+1
p:peg 3
last c
  (if c ldisk == p
   then [seq ↑[i]_p | i <- rpeg ↓[c] p]
   else
    [seq ↑[i]_(c ldisk)
       | i <- rpeg ↓[c] `p[c ldisk, p]] ++
    ↑[`c[`p[c ldisk, p]]]_p
    :: [seq ↑[i]_p | i <- ppeg `p[c ldisk, p] p]) =
`c[p]
case: eqP => [Ho|/eqP Do].
n:nat
IH:forall (c : configuration 3 n) p,
last c (rpeg c p) = `c[p]
c:configuration 3 n.+1
p:peg 3
Ho:c ldisk = p
last c [seq ↑[i]_p | i <- rpeg ↓[c] p] = `c[p]
n:nat
IH:forall (c : configuration 3 n) p,
last c (rpeg c p) = `c[p]
c:configuration 3 n.+1
p:peg 3
Do:c ldisk != p
last c
  ([seq ↑[i]_(c ldisk)
      | i <- rpeg ↓[c] `p[c ldisk, p]] ++
   ↑[`c[`p[c ldisk, p]]]_p
   :: [seq ↑[i]_p | i <- ppeg `p[c ldisk, p] p]) =
`c[p]
  by rewrite -{1}[c](cunliftrK) Ho last_map IH perfect_liftr.
n:nat
IH:forall (c : configuration 3 n) p,
last c (rpeg c p) = `c[p]
c:configuration 3 n.+1
p:peg 3
Do:c ldisk != p
last c
  ([seq ↑[i]_(c ldisk)
      | i <- rpeg ↓[c] `p[c ldisk, p]] ++
   ↑[`c[`p[c ldisk, p]]]_p
   :: [seq ↑[i]_p | i <- ppeg `p[c ldisk, p] p]) =
`c[p]
by rewrite last_cat /= last_map last_ppeg perfect_liftr.
Qed.

Lemma path_rpeg n (c : configuration 3 n) p (cs := rpeg c p) :
  path rmove c cs.
n:nat
c:configuration 3 n
p:peg 3
cs:=rpeg c p:seq (configuration 3 n)
path rmove c cs
Proof.
n:nat
c:configuration 3 n
p:peg 3
cs:=rpeg c p:seq (configuration 3 n)
path rmove c cs
have HH := @rirr 3.
n:nat
c:configuration 3 n
p:peg 3
cs:=rpeg c p:seq (configuration 3 n)
HH:irreflexive (rrel (n:=3))
path rmove c cs
rewrite /cs; elim: n c p {cs} => //= n IH c p.
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall (c : configuration 3 n) p,
path rmove c (rpeg c p)
c:configuration 3 n.+1
p:peg 3
path rmove c
  (if c ldisk == p
   then [seq ↑[i]_p | i <- rpeg ↓[c] p]
   else
    [seq ↑[i]_(c ldisk)
       | i <- rpeg ↓[c] `p[c ldisk, p]] ++
    ↑[`c[`p[c ldisk, p]]]_p
    :: [seq ↑[i]_p | i <- ppeg `p[c ldisk, p] p])
case: eqP => [Ho|/eqP Do].
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall (c : configuration 3 n) p,
path rmove c (rpeg c p)
c:configuration 3 n.+1
p:peg 3
Ho:c ldisk = p
path rmove c [seq ↑[i]_p | i <- rpeg ↓[c] p]
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall (c : configuration 3 n) p,
path rmove c (rpeg c p)
c:configuration 3 n.+1
p:peg 3
Do:c ldisk != p
path rmove c
  ([seq ↑[i]_(c ldisk)
      | i <- rpeg ↓[c] `p[c ldisk, p]] ++
   ↑[`c[`p[c ldisk, p]]]_p
   :: [seq ↑[i]_p | i <- ppeg `p[c ldisk, p] p])
  by rewrite -{1}[c](cunliftrK) Ho path_liftr.
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall (c : configuration 3 n) p,
path rmove c (rpeg c p)
c:configuration 3 n.+1
p:peg 3
Do:c ldisk != p
path rmove c
  ([seq ↑[i]_(c ldisk)
      | i <- rpeg ↓[c] `p[c ldisk, p]] ++
   ↑[`c[`p[c ldisk, p]]]_p
   :: [seq ↑[i]_p | i <- ppeg `p[c ldisk, p] p])
set c2 := `c[_].
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall (c : configuration 3 n) p,
path rmove c (rpeg c p)
c:configuration 3 n.+1
p:peg 3
Do:c ldisk != p
c2:=`c[`p[c ldisk, p]]:configuration 3 n
path rmove c
  ([seq ↑[i]_(c ldisk)
      | i <- rpeg ↓[c] `p[c ldisk, p]] ++
   ↑[c2]_p
   :: [seq ↑[i]_p | i <- ppeg `p[c ldisk, p] p])
rewrite cat_path /= -{1}[c]cunliftrK !path_liftr // IH path_ppeg ?opegDr //.
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall (c : configuration 3 n) p,
path rmove c (rpeg c p)
c:configuration 3 n.+1
p:peg 3
Do:c ldisk != p
c2:=`c[`p[c ldisk, p]]:configuration 3 n
[&& true,
    last c
      [seq ↑[i]_(c ldisk)
         | i <- rpeg ↓[c] `p[c ldisk, p]] `-->_r ↑[c2]_p
  & true]
rewrite andbT /=.
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall (c : configuration 3 n) p,
path rmove c (rpeg c p)
c:configuration 3 n.+1
p:peg 3
Do:c ldisk != p
c2:=`c[`p[c ldisk, p]]:configuration 3 n
last c
  [seq ↑[i]_(c ldisk) | i <- rpeg ↓[c] `p[c ldisk, p]] `-->_r ↑[c2]_p
rewrite -{1}[c]cunliftrK last_map last_rpeg -/c2.
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall (c : configuration 3 n) p,
path rmove c (rpeg c p)
c:configuration 3 n.+1
p:peg 3
Do:c ldisk != p
c2:=`c[`p[c ldisk, p]]:configuration 3 n
↑[c2]_(c ldisk) `-->_r ↑[c2]_p
apply: move_liftr_perfect => //; first by rewrite eq_sym (opegDl _).
HH:irreflexive (rrel (n:=3))
n:nat
IH:forall (c : configuration 3 n) p,
path rmove c (rpeg c p)
c:configuration 3 n.+1
p:peg 3
Do:c ldisk != p
c2:=`c[`p[c ldisk, p]]:configuration 3 n
p != `p[c ldisk, p]
by rewrite eq_sym (opegDr _).
Qed.

(* We can go from any configuration to a perfect configuration *)
Lemma move_connect_rpeg n (c : configuration _ n) p : c `-->*_r `c[p].
n:nat
c:configuration 3 n
p:peg 3
c `-->*_r `c[p]
Proof.
n:nat
c:configuration 3 n
p:peg 3
c `-->*_r `c[p]
apply/connectP; exists (rpeg c p); first by apply: path_rpeg.
n:nat
c:configuration 3 n
p:peg 3
`c[p] = last c (rpeg c p)
by rewrite last_rpeg.
Qed.

(* So we can also from a perfect configuration c to any configuration *)
Lemma move_connect_lpeg n (c : configuration _ n) p : `c[p] `-->*_r c.
n:nat
c:configuration 3 n
p:peg 3
`c[p] `-->*_r c
Proof.
n:nat
c:configuration 3 n
p:peg 3
`c[p] `-->*_r c 
rewrite [connect _ _ _]connect_sym //.
n:nat
c:configuration 3 n
p:peg 3
connect (move (rrel (n:=3))) c `c[p]
n:nat
c:configuration 3 n
p:peg 3
symmetric (rrel (n:=3))
  by apply: move_connect_rpeg.
n:nat
c:configuration 3 n
p:peg 3
symmetric (rrel (n:=3))
by exact: rsym.
Qed.

(* Two configurations are always connected *)
Lemma move_connect n (c1 c2 : configuration 3 n) : c1 `-->*_r c2.
n:nat
c1, c2:configuration 3 n
c1 `-->*_r c2
Proof.
n:nat
c1, c2:configuration 3 n
c1 `-->*_r c2
by apply: connect_trans (move_connect_rpeg c1 (inord 1)) 
         (move_connect_lpeg c2 (inord 1)).
Qed.

(******************************************************************************)
(* Function that builds a path from a configuration to a peg                  *)
(******************************************************************************)


(*****************************************************************************)
(* Computes the size of rpeg                                                 *)
(*****************************************************************************)

Fixpoint size_rpeg {n : nat} : (configuration _ n) -> _ -> nat :=
  match n with
  | 0 => fun _ _ => 0
  | n1.+1 =>
      fun c p =>
        let p1 := c ldisk in
        if p1 == p then size_rpeg ↓[c] p else
        let p2 := `p[p1, p] in size_rpeg ↓[c] p2 + 2 ^ n1
  end.

Lemma size_rpegE n p (c : _ _ n) : size_rpeg c p = size (rpeg c p).
n:nat
p:peg 3
c:configuration 3 n
size_rpeg c p = size (rpeg c p)
Proof.
n:nat
p:peg 3
c:configuration 3 n
size_rpeg c p = size (rpeg c p)
elim: n p c => //= n IH p c.
n:nat
IH:forall p (c : configuration 3 n),
size_rpeg c p = size (rpeg c p)
p:peg 3
c:configuration 3 n.+1
(if c ldisk == p
 then size_rpeg ↓[c] p
 else size_rpeg ↓[c] `p[c ldisk, p] + 2 ^ n) =
size
  (if c ldisk == p
   then [seq ↑[i]_p | i <- rpeg ↓[c] p]
   else
    [seq ↑[i]_(c ldisk)
       | i <- rpeg ↓[c] `p[c ldisk, p]] ++
    ↑[`c[`p[c ldisk, p]]]_p
    :: [seq ↑[i]_p | i <- ppeg `p[c ldisk, p] p])
case: eqP => [clEp|/eqP clDp]; first by rewrite size_map IH.
n:nat
IH:forall p (c : configuration 3 n),
size_rpeg c p = size (rpeg c p)
p:peg 3
c:configuration 3 n.+1
clDp:c ldisk != p
size_rpeg ↓[c] `p[c ldisk, p] + 2 ^ n =
size
  ([seq ↑[i]_(c ldisk)
      | i <- rpeg ↓[c] `p[c ldisk, p]] ++
   ↑[`c[`p[c ldisk, p]]]_p
   :: [seq ↑[i]_p | i <- ppeg `p[c ldisk, p] p])
by rewrite size_cat /= !size_map size_ppeg prednK ?expn_gt0 // IH.
Qed.

(* Upper bound on the size *)
Lemma size_rpeg_up n (c : _ _ n) p : size_rpeg c p <= (2^ n).-1.
n:nat
c:configuration 3 n
p:peg 3
size_rpeg c p <= (2 ^ n).-1
Proof.
n:nat
c:configuration 3 n
p:peg 3
size_rpeg c p <= (2 ^ n).-1
elim: n c p => //= n IH c p.
n:nat
IH:forall (c : configuration 3 n) p,
size_rpeg c p <= (2 ^ n).-1
c:configuration 3 n.+1
p:peg 3
(if c ldisk == p
 then size_rpeg ↓[c] p
 else size_rpeg ↓[c] `p[c ldisk, p] + 2 ^ n) <=
(2 ^ n.+1).-1
case: eqP => _.
n:nat
IH:forall (c : configuration 3 n) p,
size_rpeg c p <= (2 ^ n).-1
c:configuration 3 n.+1
p:peg 3
size_rpeg ↓[c] p <= (2 ^ n.+1).-1
n:nat
IH:forall (c : configuration 3 n) p,
size_rpeg c p <= (2 ^ n).-1
c:configuration 3 n.+1
p:peg 3
size_rpeg ↓[c] `p[c ldisk, p] + 2 ^ n <= (2 ^ n.+1).-1
  apply: (leq_trans (IH _ _)).
n:nat
IH:forall (c : configuration 3 n) p,
size_rpeg c p <= (2 ^ n).-1
c:configuration 3 n.+1
p:peg 3
(2 ^ n).-1 <= (2 ^ n.+1).-1
n:nat
IH:forall (c : configuration 3 n) p,
size_rpeg c p <= (2 ^ n).-1
c:configuration 3 n.+1
p:peg 3
size_rpeg ↓[c] `p[c ldisk, p] + 2 ^ n <= (2 ^ n.+1).-1
  case: (_ ^ _) (expn_eq0 2 n) (expn_eq0 2 n.+1) 
              (leq_pexp2l (isT: 0 < 2) (ltnW (leqnn n.+1))) => //= n1 _.
n:nat
IH:forall (c : configuration 3 n) p,
size_rpeg c p <= (2 ^ n).-1
c:configuration 3 n.+1
p:peg 3
n1:nat
(2 ^ n.+1 == 0) = false ->
n1 < 2 ^ n.+1 -> n1 <= (2 ^ n.+1).-1
n:nat
IH:forall (c : configuration 3 n) p,
size_rpeg c p <= (2 ^ n).-1
c:configuration 3 n.+1
p:peg 3
size_rpeg ↓[c] `p[c ldisk, p] + 2 ^ n <= (2 ^ n.+1).-1
  by case: (_ ^ _).
n:nat
IH:forall (c : configuration 3 n) p,
size_rpeg c p <= (2 ^ n).-1
c:configuration 3 n.+1
p:peg 3
size_rpeg ↓[c] `p[c ldisk, p] + 2 ^ n <= (2 ^ n.+1).-1
apply: leq_trans (_ : (2^n).-1 + (2^n)  <= _).
n:nat
IH:forall (c : configuration 3 n) p,
size_rpeg c p <= (2 ^ n).-1
c:configuration 3 n.+1
p:peg 3
size_rpeg ↓[c] `p[c ldisk, p] + 2 ^ n <=
(2 ^ n).-1 + 2 ^ n
n:nat
IH:forall (c : configuration 3 n) p,
size_rpeg c p <= (2 ^ n).-1
c:configuration 3 n.+1
p:peg 3
(2 ^ n).-1 + 2 ^ n <= (2 ^ n.+1).-1
  by rewrite leq_add2r IH.
n:nat
IH:forall (c : configuration 3 n) p,
size_rpeg c p <= (2 ^ n).-1
c:configuration 3 n.+1
p:peg 3
(2 ^ n).-1 + 2 ^ n <= (2 ^ n.+1).-1
rewrite expnS mul2n -addnn.
n:nat
IH:forall (c : configuration 3 n) p,
size_rpeg c p <= (2 ^ n).-1
c:configuration 3 n.+1
p:peg 3
(2 ^ n).-1 + 2 ^ n <= (2 ^ n + 2 ^ n).-1
by case: (2 ^ n)  (expn_eq0 2 n)  => [|n1]; rewrite ?addn0.
Qed.

(* rpeg gives the smallest path to a perfect configuration.                   *)
(* This path is unique                                                        *)
Lemma rpeg_min n (c : configuration 3 n) p cs :
  path rmove c cs -> last c cs = `c[p] ->
  size_rpeg c p <= size cs ?= iff (cs == rpeg c p).
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
Proof.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
(* As we want this statememnt to hold for any configuration c1       *)
(* and not just for initial perfect configuration the proof is more  *)
(* intricate. We need a double  induction :                          *)
(* The first induction is used when the path has duplicates (1 case) *)
have [m sLm] := ubnP (size cs); elim: m => // m IHm in n c p cs sLm *.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
sLm:size cs < m.+1
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
(* The usual induction on the number of disks                       *)
elim : n c p cs sLm => [c1 p [|] //=|n IH c1 p cs Scs c1Pcs lc1csEp /=].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
(if c1 ldisk == p
 then size_rpeg ↓[c1] p
 else size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n) <= 
                                              size cs
                                               ?= iff 
                                              (cs ==
                                               (if
                                                 c1
                                                 ldisk ==
                                                 p
                                                then
                                                 [
                                                seq ↑[i]_p
                                                  | i <- 
                                                rpeg
                                                 ↓[c1]
                                                 p]
                                                else
                                                 [
                                                 seq ↑[i]_
                                                 (c1
                                                 ldisk)
                                                  | i <- 
                                                 rpeg
                                                 ↓[c1]
                                                 `p[
                                                 c1
                                                 ldisk, p]] ++
                                                 ↑[`c[`p[
                                                 c1
                                                 ldisk, p]]]_p
                                                 :: [
                                                 seq ↑[i]_p
                                                  | i <- 
                                                 ppeg
                                                 `p[
                                                 c1
                                                 ldisk, p]
                                                 p]))
case: (_ =P p) => [c1nEp |/eqP c1nDp].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
size_rpeg ↓[c1] p <= size cs
                  ?= iff (cs ==
                          [seq ↑[i]_p
                             | i <- rpeg ↓[c1] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
  (* the largest disk is already well-placed *)
  have lcsnEc1n : last c1 cs ldisk = c1 ldisk.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
last c1 cs ldisk = c1 ldisk
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
size_rpeg ↓[c1] p <= size cs
                  ?= iff 
                 (cs ==
                  [seq ↑[i]_p | i <- rpeg ↓[c1] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
    by rewrite lc1csEp !ffunE.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
size_rpeg ↓[c1] p <= size cs
                  ?= iff (cs ==
                          [seq ↑[i]_p
                             | i <- rpeg ↓[c1] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
  have [cs1 [c1Pcs1 lcsElcs1 /leqifP]] :=
    pathS_restrict (@rirr 3) c1Pcs.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
(if cs == [seq ↑[i]_(c1 ldisk) | i <- cs1]
 then size cs1 == size cs
 else size cs1 < size cs) ->
size_rpeg ↓[c1] p <= size cs
                  ?= iff (cs ==
                          [seq ↑[i]_p
                             | i <- rpeg ↓[c1] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
  have lcs1P : last (↓[c1]) cs1 = `c[p].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
last ↓[c1] cs1 = `c[p]
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
lcs1P:last ↓[c1] cs1 = `c[p]
(if cs == [seq ↑[i]_(c1 ldisk) | i <- cs1]
 then size cs1 == size cs
 else size cs1 < size cs) ->
size_rpeg ↓[c1] p <= size cs
                  ?= iff 
                 (cs ==
                  [seq ↑[i]_p | i <- rpeg ↓[c1] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
    by rewrite lcsElcs1 lc1csEp perfect_unliftr.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
lcs1P:last ↓[c1] cs1 = `c[p]
(if cs == [seq ↑[i]_(c1 ldisk) | i <- cs1]
 then size cs1 == size cs
 else size cs1 < size cs) ->
size_rpeg ↓[c1] p <= size cs
                  ?= iff (cs ==
                          [seq ↑[i]_p
                             | i <- rpeg ↓[c1] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
  case: eqP=> [csEcs1 /eqP<- |/eqP csDcs1 scs1L].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
lcs1P:last ↓[c1] cs1 = `c[p]
csEcs1:cs = [seq ↑[i]_(c1 ldisk) | i <- cs1]
size_rpeg ↓[c1] p <= size cs1
                  ?= iff (cs ==
                          [seq ↑[i]_p
                             | i <- rpeg ↓[c1] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
lcs1P:last ↓[c1] cs1 = `c[p]
csDcs1:cs != [seq ↑[i]_(c1 ldisk) | i <- cs1]
scs1L:size cs1 < size cs
size_rpeg ↓[c1] p <= size cs
                  ?= iff 
                 (cs ==
                  [seq ↑[i]_p | i <- rpeg ↓[c1] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
    rewrite csEcs1 c1nEp eq_map_liftr.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
lcs1P:last ↓[c1] cs1 = `c[p]
csEcs1:cs = [seq ↑[i]_(c1 ldisk) | i <- cs1]
size_rpeg ↓[c1] p <= size cs1
                  ?= iff (cs1 == rpeg ↓[c1] p)
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
lcs1P:last ↓[c1] cs1 = `c[p]
csDcs1:cs != [seq ↑[i]_(c1 ldisk) | i <- cs1]
scs1L:size cs1 < size cs
size_rpeg ↓[c1] p <= size cs
                  ?= iff 
                 (cs ==
                  [seq ↑[i]_p | i <- rpeg ↓[c1] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
    apply: IH => //.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
lcs1P:last ↓[c1] cs1 = `c[p]
csEcs1:cs = [seq ↑[i]_(c1 ldisk) | i <- cs1]
size cs1 < m.+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
lcs1P:last ↓[c1] cs1 = `c[p]
csDcs1:cs != [seq ↑[i]_(c1 ldisk) | i <- cs1]
scs1L:size cs1 < size cs
size_rpeg ↓[c1] p <= size cs
                  ?= iff 
                 (cs ==
                  [seq ↑[i]_p | i <- rpeg ↓[c1] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
    by move: Scs; rewrite csEcs1 size_map.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
lcs1P:last ↓[c1] cs1 = `c[p]
csDcs1:cs != [seq ↑[i]_(c1 ldisk) | i <- cs1]
scs1L:size cs1 < size cs
size_rpeg ↓[c1] p <= size cs
                  ?= iff (cs ==
                          [seq ↑[i]_p
                             | i <- rpeg ↓[c1] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
  apply/leqifP; case: eqP => [->|_].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
lcs1P:last ↓[c1] cs1 = `c[p]
csDcs1:cs != [seq ↑[i]_(c1 ldisk) | i <- cs1]
scs1L:size cs1 < size cs
size_rpeg ↓[c1] p ==
size [seq ↑[i]_p | i <- rpeg ↓[c1] p]
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
lcs1P:last ↓[c1] cs1 = `c[p]
csDcs1:cs != [seq ↑[i]_(c1 ldisk) | i <- cs1]
scs1L:size cs1 < size cs
size_rpeg ↓[c1] p < size cs
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
    by rewrite size_map size_rpegE.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
lcs1P:last ↓[c1] cs1 = `c[p]
csDcs1:cs != [seq ↑[i]_(c1 ldisk) | i <- cs1]
scs1L:size cs1 < size cs
size_rpeg ↓[c1] p < size cs
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
  apply: leq_ltn_trans (scs1L).
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
lcs1P:last ↓[c1] cs1 = `c[p]
csDcs1:cs != [seq ↑[i]_(c1 ldisk) | i <- cs1]
scs1L:size cs1 < size cs
size_rpeg ↓[c1] p <= size cs1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
  apply: IH => //.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nEp:c1 ldisk = p
lcsnEc1n:last c1 cs ldisk = c1 ldisk
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lcsElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
lcs1P:last ↓[c1] cs1 = `c[p]
csDcs1:cs != [seq ↑[i]_(c1 ldisk) | i <- cs1]
scs1L:size cs1 < size cs
size cs1 < m.+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
  by apply: ltn_trans Scs.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n <= size cs
                                        ?= iff (cs ==
                                                [seq ↑[i]_
                                                 (c1
                                                 ldisk)
                                                  | i <- 
                                                 rpeg
                                                 ↓[c1]
                                                 `p[
                                                 c1
                                                 ldisk, p]] ++
                                                ↑[`c[`p[
                                                c1
                                                 ldisk, p]]]_p
                                                :: [
                                                seq ↑[i]_p
                                                  | i <- 
                                                ppeg
                                                 `p[
                                                 c1
                                                 ldisk, p]
                                                 p])
pose f (c : configuration 3 n.+1) := c ldisk.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n <= size cs
                                        ?= iff (cs ==
                                                [seq ↑[i]_
                                                 (c1
                                                 ldisk)
                                                  | i <- 
                                                 rpeg
                                                 ↓[c1]
                                                 `p[
                                                 c1
                                                 ldisk, p]] ++
                                                ↑[`c[`p[
                                                c1
                                                 ldisk, p]]]_p
                                                :: [
                                                seq ↑[i]_p
                                                  | i <- 
                                                ppeg
                                                 `p[
                                                 c1
                                                 ldisk, p]
                                                 p])
have HHr := @rirr 3.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n <= size cs
                                        ?= iff (cs ==
                                                [seq ↑[i]_
                                                 (c1
                                                 ldisk)
                                                  | i <- 
                                                 rpeg
                                                 ↓[c1]
                                                 `p[
                                                 c1
                                                 ldisk, p]] ++
                                                ↑[`c[`p[
                                                c1
                                                 ldisk, p]]]_p
                                                :: [
                                                seq ↑[i]_p
                                                  | i <- 
                                                ppeg
                                                 `p[
                                                 c1
                                                 ldisk, p]
                                                 p])
have HHs := @rsym 3.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n <= size cs
                                        ?= iff (cs ==
                                                [seq ↑[i]_
                                                 (c1
                                                 ldisk)
                                                  | i <- 
                                                 rpeg
                                                 ↓[c1]
                                                 `p[
                                                 c1
                                                 ldisk, p]] ++
                                                ↑[`c[`p[
                                                c1
                                                 ldisk, p]]]_p
                                                :: [
                                                seq ↑[i]_p
                                                  | i <- 
                                                ppeg
                                                 `p[
                                                 c1
                                                 ldisk, p]
                                                 p])
(* We need to move the largest disk *)
case: path3SP c1Pcs => // [c1' cs' p1 csE c1'Pcs' _|
                           cs1 cs2 p1 p2 p3 c1' c2
                           p1Dp2 p1Rp2 lc1'cs1Epp3 csE c1'Pcs1 c2Pcs2 _].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
c1':=↓[c1]:configuration 3 n
cs':=[seq ↓[i] | i <- cs]:seq (configuration 3 n)
p1:=c1 ldisk:peg 3
csE:cs = [seq ↑[i]_p1 | i <- cs']
c1'Pcs':path (move (rrel (n:=3))) c1' cs'
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n <= size cs
                                        ?= iff (cs ==
                                                [seq ↑[i]_
                                                 (c1
                                                 ldisk)
                                                  | i <- 
                                                 rpeg
                                                 ↓[c1]
                                                 `p[
                                                 c1
                                                 ldisk, p]] ++
                                                ↑[`c[`p[
                                                c1
                                                 ldisk, p]]]_p
                                                :: [
                                                seq ↑[i]_p
                                                  | i <- 
                                                ppeg
                                                 `p[
                                                 c1
                                                 ldisk, p]
                                                 p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
  (* this case is impossible the largest disk has to move *)
  case/eqP: c1nDp.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
c1':=↓[c1]:configuration 3 n
cs':=[seq ↓[i] | i <- cs]:seq (configuration 3 n)
p1:=c1 ldisk:peg 3
csE:cs = [seq ↑[i]_p1 | i <- cs']
c1'Pcs':path (move (rrel (n:=3))) c1' cs'
c1 ldisk = p
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
  move : lc1csEp; rewrite csE -[c1](cunliftrK) last_map => /(congr1 f).
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
c1':=↓[c1]:configuration 3 n
cs':=[seq ↓[i] | i <- cs]:seq (configuration 3 n)
p1:=c1 ldisk:peg 3
csE:cs = [seq ↑[i]_p1 | i <- cs']
c1'Pcs':path (move (rrel (n:=3))) c1' cs'
f ↑[last ↓[c1] cs']_(c1 ldisk) = f `c[p] ->
↑[↓[c1]]_(c1 ldisk) ldisk = p
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n
 <= size cs
 ?= iff (cs ==
         [seq ↑[i]_(c1 ldisk)
            | i <- rpeg ↓[c1] `p[c1 ldisk, p]] ++
         ↑[`c[`p[c1 ldisk, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[c1 ldisk, p] p])
  by rewrite /f !cliftr_ldisk !ffunE.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n <= size cs
                                        ?= iff (cs ==
                                                [seq ↑[i]_
                                                 (c1
                                                 ldisk)
                                                  | i <- 
                                                 rpeg
                                                 ↓[c1]
                                                 `p[
                                                 c1
                                                 ldisk, p]] ++
                                                ↑[`c[`p[
                                                c1
                                                 ldisk, p]]]_p
                                                :: [
                                                seq ↑[i]_p
                                                  | i <- 
                                                ppeg
                                                 `p[
                                                 c1
                                                 ldisk, p]
                                                 p])
(* c2 is the first configuration when the largest disk has moved *)
rewrite csE size_cat -/p1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
have p1Dp : p1 != p by [].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
have Scs1 : size cs1 < m.+1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
size cs1 < m.+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  apply: ltn_trans Scs.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
size cs1 < size cs
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  by rewrite csE size_cat /= size_map addnS ltnS leq_addr.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
have Scs2 : size cs2 < m.+1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
size cs2 < m.+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  apply: ltn_trans Scs.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
size cs2 < size cs
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  by rewrite csE size_cat /= size_map addnS ltnS leq_addl.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
case: (p2 =P p) => [p2Ep|/eqP p2Dp].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  pose c2' := ↓[c2].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  have c2'Epp3 : c2' = `c[`p[p1, p2]] by rewrite [LHS]cliftrK.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  (* the first moves of largest disk of cs is the right one *)
  have/(pathS_restrict HHr)[cs2' [c2'Pcs2' lc2'cs2'E cs2'L]] := c2Pcs2.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
cs2'L:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c2 ldisk)
                     | i <- cs2'])
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  have Scs2' : size cs2' < m.+1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
cs2'L:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c2 ldisk)
                     | i <- cs2'])
size cs2' < m.+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
cs2'L:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c2 ldisk)
                     | i <- cs2'])
Scs2':size cs2' < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
    by apply: leq_ltn_trans Scs2; rewrite cs2'L.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
cs2'L:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c2 ldisk)
                     | i <- cs2'])
Scs2':size cs2' < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  have /IH := lc1'cs1Epp3 => // /(_ Scs1 c1'Pcs1) IHc1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
cs2'L:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c2 ldisk)
                     | i <- cs2'])
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  have /IH : last c2' cs2' = `c[p] => [| /(_ Scs2' c2'Pcs2') IHc2].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
cs2'L:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c2 ldisk)
                     | i <- cs2'])
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
last c2' cs2' = `c[p]
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
cs2'L:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c2 ldisk)
                     | i <- cs2'])
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p <= size cs2'
                ?= iff 
               (cs2' == rpeg c2' p)
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
    rewrite lc2'cs2'E -perfect_unliftr.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
cs2'L:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c2 ldisk)
                     | i <- cs2'])
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
↓[last c2 cs2] = ↓[`c[p]]
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
cs2'L:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c2 ldisk)
                     | i <- cs2'])
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p <= size cs2'
                ?= iff 
               (cs2' == rpeg c2' p)
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
    by move: lc1csEp; rewrite csE last_cat /= => ->.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
cs2'L:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c2 ldisk)
                     | i <- cs2'])
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p <= size cs2'
                ?= iff 
               (cs2' == rpeg c2' p)
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  rewrite -p2Ep in IHc2.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
cs2'L:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c2 ldisk)
                     | i <- cs2'])
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  move /leqifP : cs2'L.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
(if cs2 == [seq ↑[i]_(c2 ldisk) | i <- cs2']
 then size cs2' == size cs2
 else size cs2' < size cs2) ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  case: eqP => [cs2E _ | /eqP cs2D Lcs2].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
    (* there is only one move of the largest disk in cs *)
    rewrite cs2E size_map -p2Ep -/p3.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
size_rpeg ↓[c1] p3 + 2 ^ n <= size cs1 +
                              size
                                (c2
                                 :: [seq ↑[i]_(c2
                                                 ldisk)
                                       | i <- cs2'])
                           ?= iff ([seq ↑[i]_p1
                                      | i <- cs1] ++
                                   c2
                                   :: [seq ↑[i]_(c2
                                                 ldisk)
                                         | i <- cs2'] ==
                                   [seq ↑[i]_p1
                                      | i <- rpeg
                                               ↓[c1]
                                               p3] ++
                                   ↑[`c[p3]]_p2
                                   :: [seq ↑[i]_p2
                                         | i <- ppeg
                                                 p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
    have /leqifP := IHc1; case: eqP => [->_ |_ Lc1].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
size_rpeg ↓[c1] p3 + 2 ^ n <= size (rpeg c1' p3) +
                              size
                                (c2
                                 :: [seq ↑[i]_(c2
                                                 ldisk)
                                       | i <- cs2'])
                           ?= iff ([seq ↑[i]_p1
                                      | i <- rpeg c1'
                                               p3] ++
                                   c2
                                   :: [seq ↑[i]_(c2
                                                 ldisk)
                                         | i <- cs2'] ==
                                   [seq ↑[i]_p1
                                      | i <- rpeg
                                               ↓[c1]
                                               p3] ++
                                   ↑[`c[p3]]_p2
                                   :: [seq ↑[i]_p2
                                         | i <- ppeg
                                                 p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size cs1 +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
      (* the first part of cs is perfect *)
      have /leqifP := IHc2; case: eqP => [->_ |_ Lc2].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
size_rpeg ↓[c1] p3 + 2 ^ n <= size (rpeg c1' p3) +
                              size
                                (c2
                                 :: [seq ↑[i]_(c2
                                                 ldisk)
                                       | i <- rpeg c2'
                                                p2])
                           ?= iff ([seq ↑[i]_p1
                                      | i <- rpeg c1'
                                               p3] ++
                                   c2
                                   :: [seq ↑[i]_(c2
                                                 ldisk)
                                         | i <- rpeg
                                                 c2'
                                                 p2] ==
                                   [seq ↑[i]_p1
                                      | i <- rpeg
                                               ↓[c1]
                                               p3] ++
                                   ↑[`c[p3]]_p2
                                   :: [seq ↑[i]_p2
                                         | i <- ppeg
                                                 p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc2:size_rpeg c2' p2 < size cs2'
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size (rpeg c1' p3) +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- rpeg c1' p3] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size cs1 +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
      (* the second part of cs is perfect, only case of equality *)
        apply/leqifP; case: eqP => [/(congr1 size)|[]].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
size
  ([seq ↑[i]_p1 | i <- rpeg c1' p3] ++
   c2 :: [seq ↑[i]_(c2 ldisk) | i <- rpeg c2' p2]) =
size
  ([seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
   ↑[`c[p3]]_p2 :: [seq ↑[i]_p2 | i <- ppeg p3 p2]) ->
size_rpeg ↓[c1] p3 + 2 ^ n ==
size (rpeg c1' p3) +
size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- rpeg c2' p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
[seq ↑[i]_p1 | i <- rpeg c1' p3] ++
c2 :: [seq ↑[i]_(c2 ldisk) | i <- rpeg c2' p2] =
[seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
↑[`c[p3]]_p2 :: [seq ↑[i]_p2 | i <- ppeg p3 p2]
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc2:size_rpeg c2' p2 < size cs2'
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size (rpeg c1' p3) +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- rpeg c1' p3] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size cs1 +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
          rewrite !size_cat /= !size_map !addnS !size_rpegE c2'Epp3.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
(size (rpeg c1' p3) + size (rpeg `c[`p[p1, p2]] p2)).+1 =
(size (rpeg c1' p3) + size (ppeg p3 p2)).+1 ->
size (rpeg ↓[c1] p3) + 2 ^ n ==
(size (rpeg c1' p3) + size (rpeg `c[`p[p1, p2]] p2)).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
[seq ↑[i]_p1 | i <- rpeg c1' p3] ++
c2 :: [seq ↑[i]_(c2 ldisk) | i <- rpeg c2' p2] =
[seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
↑[`c[p3]]_p2 :: [seq ↑[i]_p2 | i <- ppeg p3 p2]
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc2:size_rpeg c2' p2 < size cs2'
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size (rpeg c1' p3) +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- rpeg c1' p3] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size cs1 +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
          by rewrite !rpeg_ppeg ?opegDr // size_ppeg -addnS prednK ?expn_gt0.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
[seq ↑[i]_p1 | i <- rpeg c1' p3] ++
c2 :: [seq ↑[i]_(c2 ldisk) | i <- rpeg c2' p2] =
[seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
↑[`c[p3]]_p2 :: [seq ↑[i]_p2 | i <- ppeg p3 p2]
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc2:size_rpeg c2' p2 < size cs2'
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size (rpeg c1' p3) +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- rpeg c1' p3] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size cs1 +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
        congr (_ ++ _ :: _).
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
[seq ↑[i]_(c2 ldisk) | i <- rpeg c2' p2] =
[seq ↑[i]_p2 | i <- ppeg p3 p2]
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc2:size_rpeg c2' p2 < size cs2'
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size (rpeg c1' p3) +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- rpeg c1' p3] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size cs1 +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
        by rewrite /c2' /c2 cliftrK cliftr_ldisk rpeg_ppeg // opegDr.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc2:size_rpeg c2' p2 < size cs2'
size_rpeg ↓[c1] p3 + 2 ^ n <= size (rpeg c1' p3) +
                              size
                                (c2
                                 :: [seq ↑[i]_(c2
                                                 ldisk)
                                       | i <- cs2'])
                           ?= iff ([seq ↑[i]_p1
                                      | i <- rpeg c1'
                                               p3] ++
                                   c2
                                   :: [seq ↑[i]_(c2
                                                 ldisk)
                                         | i <- cs2'] ==
                                   [seq ↑[i]_p1
                                      | i <- rpeg
                                               ↓[c1]
                                               p3] ++
                                   ↑[`c[p3]]_p2
                                   :: [seq ↑[i]_p2
                                         | i <- ppeg
                                                 p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size cs1 +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
      apply/leqifP; case: eqP => [/(congr1 size)|_].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc2:size_rpeg c2' p2 < size cs2'
size
  ([seq ↑[i]_p1 | i <- rpeg c1' p3] ++
   c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2']) =
size
  ([seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
   ↑[`c[p3]]_p2 :: [seq ↑[i]_p2 | i <- ppeg p3 p2]) ->
size_rpeg ↓[c1] p3 + 2 ^ n ==
size (rpeg c1' p3) +
size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc2:size_rpeg c2' p2 < size cs2'
size_rpeg ↓[c1] p3 + 2 ^ n <
size (rpeg c1' p3) +
size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size cs1 +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
        rewrite !size_cat /= !size_map !size_rpegE.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc2:size_rpeg c2' p2 < size cs2'
size (rpeg c1' p3) + (size cs2').+1 =
size (rpeg c1' p3) + (size (ppeg p3 p2)).+1 ->
size (rpeg ↓[c1] p3) + 2 ^ n ==
size (rpeg c1' p3) + (size cs2').+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc2:size_rpeg c2' p2 < size cs2'
size_rpeg ↓[c1] p3 + 2 ^ n <
size (rpeg c1' p3) +
size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size cs1 +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
        by rewrite size_ppeg prednK ?expn_gt0 // => ->.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc2:size_rpeg c2' p2 < size cs2'
size_rpeg ↓[c1] p3 + 2 ^ n <
size (rpeg c1' p3) +
size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size cs1 +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
      rewrite /= size_rpegE ltn_add2l ltnS size_map.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc2:size_rpeg c2' p2 < size cs2'
2 ^ n <= size cs2'
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size cs1 +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
      rewrite c2'Epp3 size_rpegE rpeg_ppeg ?opegDr // size_ppeg in Lc2.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc2:(2 ^ n).-1 < size cs2'
2 ^ n <= size cs2'
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n
 <= size cs1 +
    size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++
         c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'] ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
         ↑[`c[p3]]_p2
         :: [seq ↑[i]_p2 | i <- ppeg p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
      by rewrite prednK // expn_gt0 in Lc2.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n <= size cs1 +
                              size
                                (c2
                                 :: [seq ↑[i]_(c2
                                                 ldisk)
                                       | i <- cs2'])
                           ?= iff ([seq ↑[i]_p1
                                      | i <- cs1] ++
                                   c2
                                   :: [seq ↑[i]_(c2
                                                 ldisk)
                                         | i <- cs2'] ==
                                   [seq ↑[i]_p1
                                      | i <- rpeg
                                               ↓[c1]
                                               p3] ++
                                   ↑[`c[p3]]_p2
                                   :: [seq ↑[i]_p2
                                         | i <- ppeg
                                                 p3 p2])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
    apply/leqifP; case: eqP => [/(congr1 size)|_].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size
  ([seq ↑[i]_p1 | i <- cs1] ++
   c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2']) =
size
  ([seq ↑[i]_p1 | i <- rpeg ↓[c1] p3] ++
   ↑[`c[p3]]_p2 :: [seq ↑[i]_p2 | i <- ppeg p3 p2]) ->
size_rpeg ↓[c1] p3 + 2 ^ n ==
size cs1 +
size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n <
size cs1 +
size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
      rewrite !size_cat /= !size_map !size_rpegE => ->.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size (rpeg ↓[c1] p3) + 2 ^ n ==
size (rpeg ↓[c1] p3) + (size (ppeg p3 p2)).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n <
size cs1 +
size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
      by rewrite size_ppeg prednK // expn_gt0.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
size_rpeg ↓[c1] p3 + 2 ^ n <
size cs1 +
size (c2 :: [seq ↑[i]_(c2 ldisk) | i <- cs2'])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
    rewrite /= -addSn leq_add // size_map -[_ ^ _]prednK ?expn_gt0 // ltnS.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
(2 ^ n).-1 <= size cs2'
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
    have := IHc2; rewrite c2'Epp3 size_rpegE rpeg_ppeg ?opegDr // size_ppeg.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2E:cs2 = [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lc1:size_rpeg c1' p3 < size cs1
(2 ^ n).-1 <= size cs2'
           ?= iff (cs2' == ppeg `p[p1, p2] p2) ->
(2 ^ n).-1 <= size cs2'
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
    by move=>->.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  apply/leqifP; case: eqP => [/(congr1 size)|_].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2) =
size
  ([seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
   ↑[`c[`p[p1, p]]]_p
   :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p]) ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n ==
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
    rewrite !size_cat /= !size_map !size_rpegE.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size cs1 + (size cs2).+1 =
size (rpeg ↓[c1] `p[p1, p]) +
(size (ppeg `p[p1, p] p)).+1 ->
size (rpeg ↓[c1] `p[p1, p]) + 2 ^ n ==
size cs1 + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
    by rewrite size_ppeg prednK ?expn_gt0 // => ->.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  rewrite -addnS leq_add //= ?ltnS -?p2Ep.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
size_rpeg ↓[c1] `p[p1, p2] <=
size [seq ↑[i]_p1 | i <- cs1]
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
2 ^ n <= size cs2
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
    by rewrite size_map IHc1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
2 ^ n <= size cs2
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  rewrite -[_ ^ _]prednK ?expn_gt0 //.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
Lcs2:size cs2' < size cs2
(2 ^ n).-1 < size cs2
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  apply: leq_ltn_trans Lcs2.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
(2 ^ n).-1 <= size cs2'
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  have := IHc2.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Ep:p2 = p
c2':=↓[c2]:configuration 3 n
c2'Epp3:c2' = `c[`p[p1, p2]]
cs2':seq (configuration 3 n)
c2'Pcs2':path (move (rrel (n:=3))) ↓[c2] cs2'
lc2'cs2'E:last ↓[c2] cs2' = ↓[last c2 cs2]
Scs2':size cs2' < m.+1
IHc1:size_rpeg c1' p3 <= size cs1
                 ?= iff 
                (cs1 == rpeg c1' p3)
IHc2:size_rpeg c2' p2 <= size cs2'
                 ?= iff 
                (cs2' == rpeg c2' p2)
cs2D:cs2 != [seq ↑[i]_(c2 ldisk) | i <- cs2']
size_rpeg c2' p2 <= size cs2'
                 ?= iff (cs2' == rpeg c2' p2) ->
(2 ^ n).-1 <= size cs2'
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  by rewrite c2'Epp3 size_rpegE rpeg_ppeg ?opegDr // size_ppeg => ->.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
(* The largest disk jumped to an intermediate peg *)
have p3Ep : p3 = p.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3 = p
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n
 <= size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
 ?= iff ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2 ==
         [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
         ↑[`c[`p[p1, p]]]_p
         :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p])
  by apply/eqP; rewrite opeg3E // p1Dp.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size
                                       [seq ↑[i]_p1
                                          | i <- cs1] +
                                     size (c2 :: cs2)
                                  ?= iff ([seq ↑[i]_p1
                                             | i <- cs1] ++
                                          c2 :: cs2 ==
                                          [seq ↑[i]_p1
                                             | i <- 
                                           rpeg ↓[c1]
                                             `p[p1, p]] ++
                                          ↑[`c[`p[p1, p]]]_p
                                          :: [seq ↑[i]_p
                                                | i <- 
                                              ppeg
                                                `p[p1, p]
                                                p])
(* cs cannot be optimal *)
apply/leqifP; case: eqP => [/(congr1 size)|_].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
size ([seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2) =
size
  ([seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p]] ++
   ↑[`c[`p[p1, p]]]_p
   :: [seq ↑[i]_p | i <- ppeg `p[p1, p] p]) ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n ==
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  rewrite !size_cat /= !size_map !size_rpegE.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
size cs1 + (size cs2).+1 =
size (rpeg ↓[c1] `p[p1, p]) +
(size (ppeg `p[p1, p] p)).+1 ->
size (rpeg ↓[c1] `p[p1, p]) + 2 ^ n ==
size cs1 + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  by rewrite size_ppeg prednK ?expn_gt0 // => ->.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c2Pcs2:path (move (rrel (n:=3))) c2 cs2
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
case: path3SP c2Pcs2 => // [c2' cs2'|cs3 cs4].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
c2':=↓[c2]:configuration 3 n
cs2':=[seq ↓[i] | i <- cs2]:seq (configuration 3 n)
let p0 := c2 ldisk in
cs2 = [seq ↑[i]_p0 | i <- cs2'] ->
path (move (rrel (n:=3))) c2' cs2' ->
true ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i0]_p1 | i0 <- cs1] + size (c2 :: cs2)
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
let p2 := c2 ldisk in
forall p3,
let p4 := `p[p2, p3] in
let c3 := ↓[c2] in
let c4 := ↑[`c[p4]]_p3 in
p2 != p3 ->
rrel p2 p3 ->
last c3 cs3 = `c[p4] ->
cs2 = [seq ↑[i1]_p2 | i1 <- cs3] ++ c4 :: cs4 ->
path (move (rrel (n:=3))) c3 cs3 ->
path (move (rrel (n:=3))) c4 cs4 ->
true ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i3]_p1 | i3 <- cs1] + size (c2 :: cs2)
  (* this is impossible we need another move of the largest disk *)  
  rewrite !cliftr_ldisk /= => cs2E c2'Pcs2'.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
c2':=↓[c2]:configuration 3 n
cs2':=[seq ↓[i] | i <- cs2]:seq (configuration 3 n)
cs2E:cs2 = [seq ↑[i]_p2 | i <- cs2']
c2'Pcs2':path (move (rrel (n:=3))) c2' cs2'
true ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
let p2 := c2 ldisk in
forall p3,
let p4 := `p[p2, p3] in
let c3 := ↓[c2] in
let c4 := ↑[`c[p4]]_p3 in
p2 != p3 ->
rrel p2 p3 ->
last c3 cs3 = `c[p4] ->
cs2 = [seq ↑[i1]_p2 | i1 <- cs3] ++ c4 :: cs4 ->
path (move (rrel (n:=3))) c3 cs3 ->
path (move (rrel (n:=3))) c4 cs4 ->
true ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i3]_p1 | i3 <- cs1] + size (c2 :: cs2)
  case/eqP: p2Dp.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p3Ep:p3 = p
c2':=↓[c2]:configuration 3 n
cs2':=[seq ↓[i] | i <- cs2]:seq (configuration 3 n)
cs2E:cs2 = [seq ↑[i]_p2 | i <- cs2']
c2'Pcs2':path (move (rrel (n:=3))) c2' cs2'
p2 = p
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
let p2 := c2 ldisk in
forall p3,
let p4 := `p[p2, p3] in
let c3 := ↓[c2] in
let c4 := ↑[`c[p4]]_p3 in
p2 != p3 ->
rrel p2 p3 ->
last c3 cs3 = `c[p4] ->
cs2 = [seq ↑[i1]_p2 | i1 <- cs3] ++ c4 :: cs4 ->
path (move (rrel (n:=3))) c3 cs3 ->
path (move (rrel (n:=3))) c4 cs4 ->
true ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i3]_p1 | i3 <- cs1] + size (c2 :: cs2)
  have := lc1csEp.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p3Ep:p3 = p
c2':=↓[c2]:configuration 3 n
cs2':=[seq ↓[i] | i <- cs2]:seq (configuration 3 n)
cs2E:cs2 = [seq ↑[i]_p2 | i <- cs2']
c2'Pcs2':path (move (rrel (n:=3))) c2' cs2'
last c1 cs = `c[p] -> p2 = p
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
let p2 := c2 ldisk in
forall p3,
let p4 := `p[p2, p3] in
let c3 := ↓[c2] in
let c4 := ↑[`c[p4]]_p3 in
p2 != p3 ->
rrel p2 p3 ->
last c3 cs3 = `c[p4] ->
cs2 = [seq ↑[i1]_p2 | i1 <- cs3] ++ c4 :: cs4 ->
path (move (rrel (n:=3))) c3 cs3 ->
path (move (rrel (n:=3))) c4 cs4 ->
true ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i3]_p1 | i3 <- cs1] + size (c2 :: cs2)
  rewrite csE last_cat /= => /(congr1 f).
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p3Ep:p3 = p
c2':=↓[c2]:configuration 3 n
cs2':=[seq ↓[i] | i <- cs2]:seq (configuration 3 n)
cs2E:cs2 = [seq ↑[i]_p2 | i <- cs2']
c2'Pcs2':path (move (rrel (n:=3))) c2' cs2'
f (last c2 cs2) = f `c[p] -> p2 = p
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
let p2 := c2 ldisk in
forall p3,
let p4 := `p[p2, p3] in
let c3 := ↓[c2] in
let c4 := ↑[`c[p4]]_p3 in
p2 != p3 ->
rrel p2 p3 ->
last c3 cs3 = `c[p4] ->
cs2 = [seq ↑[i1]_p2 | i1 <- cs3] ++ c4 :: cs4 ->
path (move (rrel (n:=3))) c3 cs3 ->
path (move (rrel (n:=3))) c4 cs4 ->
true ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i3]_p1 | i3 <- cs1] + size (c2 :: cs2)
  by rewrite /f cs2E last_map !cliftr_ldisk !ffunE.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
let p2 := c2 ldisk in
forall p3,
let p4 := `p[p2, p3] in
let c3 := ↓[c2] in
let c4 := ↑[`c[p4]]_p3 in
p2 != p3 ->
rrel p2 p3 ->
last c3 cs3 = `c[p4] ->
cs2 = [seq ↑[i1]_p2 | i1 <- cs3] ++ c4 :: cs4 ->
path (move (rrel (n:=3))) c3 cs3 ->
path (move (rrel (n:=3))) c4 cs4 ->
true ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i3]_p1 | i3 <- cs1] + size (c2 :: cs2)
rewrite !cliftr_ldisk => p2'.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p2':=p2:peg 3
forall p2,
let p3 := `p[p2', p2] in
let c3 := ↓[c2] in
let c4 := ↑[`c[p3]]_p2 in
p2' != p2 ->
rrel p2' p2 ->
last c3 cs3 = `c[p3] ->
cs2 = [seq ↑[i1]_p2' | i1 <- cs3] ++ c4 :: cs4 ->
path (move (rrel (n:=3))) c3 cs3 ->
path (move (rrel (n:=3))) c4 cs4 ->
true ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i3]_p1 | i3 <- cs1] + size (c2 :: cs2)
rewrite {}/p2' => p4.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p2 != p4 ->
rrel p2 p4 ->
last ↓[c2] cs3 = `c[`p[p2, p4]] ->
cs2 =
[seq ↑[i1]_p2 | i1 <- cs3] ++
↑[`c[`p[p2, p4]]]_p4 :: cs4 ->
path (move (rrel (n:=3))) ↓[c2] cs3 ->
path (move (rrel (n:=3))) ↑[`c[`p[p2, p4]]]_p4 cs4 ->
true ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i3]_p1 | i3 <- cs1] + size (c2 :: cs2)
set p5 := `p[p2, p4]; set c2' := ↓[c2].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2 != p4 ->
rrel p2 p4 ->
last c2' cs3 = `c[p5] ->
cs2 =
[seq ↑[i1]_p2 | i1 <- cs3] ++ ↑[`c[p5]]_p4 :: cs4 ->
path (move (rrel (n:=3))) c2' cs3 ->
path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4 ->
true ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i3]_p1 | i3 <- cs1] + size (c2 :: cs2)
move=> p2Dp4 p2Rp4 c2'cs5Epp5 cs2E c2'Pcs3 pp5p4Pcs4 _.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
case: (p5 =P p3) => [p5Ep3 /= | /eqP p5Dp3].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  (* the path has a duplicate use the induction hypothesis *)
  have p4Ep1: p4 = p1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4 = p1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
    move/eqP : p5Ep3; rewrite eq_sym // opeg3E // eq_sym //.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
(p5 != p1) && (p2 != p5) -> p4 = p1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
    by rewrite opeg3E  // negb_and !negbK eq_sym (negPf p1Dp2) 
          => /andP[/eqP] .
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  pose cs5 := [seq ↑[i]_p1 | i <- cs1] ++ cs4.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  have scs5Lscs : size cs5 < size cs.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
size cs5 < size cs
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
    rewrite /cs5 csE cs2E !size_cat /= !size_cat /= !size_map.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
size cs1 + size cs4 <
size cs1 + (size cs3 + (size cs4).+1).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
    rewrite ltn_add2l // !addnS! ltnS -addSn leq_addl //.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  have c1Mcs5 : path rmove c1 cs5.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
path rmove c1 cs5
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
    rewrite cat_path -[c1](cunliftrK) !path_liftr //=.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
path (move (rrel (n:=3))) ↓[c1] cs1 &&
path rmove
  (last ↑[↓[c1]]_(c1 ldisk) [seq ↑[i]_p1 | i <- cs1])
  cs4
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
    rewrite c1'Pcs1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
true &&
path rmove
  (last ↑[↓[c1]]_(c1 ldisk) [seq ↑[i]_p1 | i <- cs1])
  cs4
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
    by rewrite last_map lc1'cs1Epp3 -/p1 -p5Ep3 -p4Ep1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  have lc1cs5E : last c1 cs5 = `c[p].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
last c1 cs5 = `c[p]
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
lc1cs5E:last c1 cs5 = `c[p]
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
    rewrite last_cat -[c1](cunliftrK) last_map lc1'cs1Epp3.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
last ↑[`c[p3]]_(c1 ldisk) cs4 = `c[p]
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
lc1cs5E:last c1 cs5 = `c[p]
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
    rewrite -lc1csEp csE cs2E last_cat /= last_cat /= -/p1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
last ↑[`c[p3]]_p1 cs4 = last ↑[`c[p5]]_p4 cs4
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
lc1cs5E:last c1 cs5 = `c[p]
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
    by rewrite p5Ep3 p4Ep1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
lc1cs5E:last c1 cs5 = `c[p]
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  apply: leq_trans (_ : size cs5 < _); last first.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
lc1cs5E:last c1 cs5 = `c[p]
size cs5 <
size [seq ↑[i]_p1 | i <- cs1] + (size cs2).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
lc1cs5E:last c1 cs5 = `c[p]
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n < (size cs5).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
    by rewrite cs2E /= !size_cat !size_map /= 
               ltn_add2l // ltnS -addSnnS leq_addl.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
lc1cs5E:last c1 cs5 = `c[p]
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n < (size cs5).+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  rewrite ltnS.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
lc1cs5E:last c1 cs5 = `c[p]
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size cs5
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)    
  have /IHm : size cs5 < m.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
lc1cs5E:last c1 cs5 = `c[p]
size cs5 < m
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
lc1cs5E:last c1 cs5 = `c[p]
(forall (c : configuration 3 n.+1) p,
 path rmove c cs5 ->
 last c cs5 = `c[p] ->
 size_rpeg c p <= size cs5 ?= iff (cs5 == rpeg c p)) ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size cs5
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
    rewrite -ltnS.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
lc1cs5E:last c1 cs5 = `c[p]
(size cs5).+1 < m.+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
lc1cs5E:last c1 cs5 = `c[p]
(forall (c : configuration 3 n.+1) p,
 path rmove c cs5 ->
 last c cs5 = `c[p] ->
 size_rpeg c p <= size cs5 ?= iff (cs5 == rpeg c p)) ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size cs5
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
    by apply: leq_ltn_trans Scs.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
lc1cs5E:last c1 cs5 = `c[p]
(forall (c : configuration 3 n.+1) p,
 path rmove c cs5 ->
 last c cs5 = `c[p] ->
 size_rpeg c p <= size cs5 ?= iff (cs5 == rpeg c p)) ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size cs5
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  move=> /(_ c1 p c1Mcs5 lc1cs5E) /=.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Ep3:p5 = p3
p4Ep1:p4 = p1
cs5:=[seq ↑[i]_p1 | i <- cs1] ++ cs4:seq (configuration 3 n.+1)
scs5Lscs:size cs5 < size cs
c1Mcs5:path rmove c1 cs5
lc1cs5E:last c1 cs5 = `c[p]
(if c1 ldisk == p
 then size_rpeg ↓[c1] p
 else size_rpeg ↓[c1] `p[c1 ldisk, p] + 2 ^ n) <= 
                                              size cs5
                                               ?= iff 
                                              (cs5 ==
                                               (if
                                                 c1
                                                 ldisk ==
                                                 p
                                                then
                                                 [
                                                seq ↑[i]_p
                                                  | i <- 
                                                rpeg
                                                 ↓[c1]
                                                 p]
                                                else
                                                 [
                                                 seq ↑[i]_
                                                 (c1
                                                 ldisk)
                                                  | i <- 
                                                 rpeg
                                                 ↓[c1]
                                                 `p[
                                                 c1
                                                 ldisk, p]] ++
                                                 ↑[`c[`p[
                                                 c1
                                                 ldisk, p]]]_p
                                                 :: [
                                                 seq ↑[i]_p
                                                  | i <- 
                                                 ppeg
                                                 `p[
                                                 c1
                                                 ldisk, p]
                                                 p])) ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <= size cs5
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  by rewrite -/p1 (negPf p1Dp) => ->.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
(* now we just need to use the induction principle on the two subpath        *)
have [cs4' [pp5p4Pcs4' lpp5p4cs4'Elpp5p4cs4 scs4'L]] := 
  pathS_restrict HHr pp5p4Pcs4.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
pp5p4Pcs4':path (move (rrel (n:=3))) ↓[↑[`c[p5]]_p4]
  cs4'
lpp5p4cs4'Elpp5p4cs4:last ↓[↑[`c[p5]]_p4] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
rewrite cliftrK in lpp5p4cs4'Elpp5p4cs4 pp5p4Pcs4'.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
have Scs3 : size cs3 < m.+1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
size cs3 < m.+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  apply: ltn_trans Scs2.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
size cs3 < size cs2
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  by rewrite cs2E size_cat /= size_map addnS ltnS leq_addr.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
have Scs4 : size cs4 < m.+1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
size cs4 < m.+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  apply: ltn_trans Scs2.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
size cs4 < size cs2
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  by rewrite cs2E size_cat /= size_map addnS ltnS leq_addl.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
have Scs4' : size cs4' < m.+1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
size cs4' < m.+1
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
  by apply: leq_ltn_trans Scs4; rewrite scs4'L.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size [seq ↑[i]_p1 | i <- cs1] + size (c2 :: cs2)
rewrite cs2E /= size_cat /= !size_map.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size cs1 + (size cs3 + (size cs4).+1).+1
have /IH := c2'cs5Epp5 => /(_ Scs3 c2'Pcs3) IHc1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
IHc1:size_rpeg c2' p5 <= size cs3
                 ?= iff 
                (cs3 == rpeg c2' p5)
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size cs1 + (size cs3 + (size cs4).+1).+1
have c2'E : c2' = `c[p3] by rewrite [LHS]cliftrK.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
IHc1:size_rpeg c2' p5 <= size cs3
                 ?= iff 
                (cs3 == rpeg c2' p5)
c2'E:c2' = `c[p3]
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size cs1 + (size cs3 + (size cs4).+1).+1
rewrite c2'E size_rpegE rpeg_ppeg // in IHc1; last by rewrite eq_sym.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:size (ppeg p3 p5) <= size cs3
                  ?= iff 
                 (cs3 == ppeg p3 p5)
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size cs1 + (size cs3 + (size cs4).+1).+1
rewrite size_ppeg in IHc1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size cs1 + (size cs3 + (size cs4).+1).+1
have p6Dp : p5 != p.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p5 != p
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size cs1 + (size cs3 + (size cs4).+1).+1
  by apply: contra p5Dp3 => /eqP->; apply/eqP.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
lc1csEp:last c1 cs = `c[p]
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size cs1 + (size cs3 + (size cs4).+1).+1
move: lc1csEp; rewrite csE cs2E.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
last c1
  ([seq ↑[i]_p1 | i <- cs1] ++
   c2
   :: [seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4) =
`c[p] ->
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size cs1 + (size cs3 + (size cs4).+1).+1
rewrite last_cat /= last_cat /= => lpp5p4cs4Epp.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
lpp5p4cs4Epp:last ↑[`c[p5]]_p4 cs4 = `c[p]
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size cs1 + (size cs3 + (size cs4).+1).+1
have lpp5cs4'Epp : last `c[p5] cs4' = `c[p].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
lpp5p4cs4Epp:last ↑[`c[p5]]_p4 cs4 = `c[p]
last `c[p5] cs4' = `c[p]
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
lpp5p4cs4Epp:last ↑[`c[p5]]_p4 cs4 = `c[p]
lpp5cs4'Epp:last `c[p5] cs4' = `c[p]
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size cs1 + (size cs3 + (size cs4).+1).+1
  by rewrite lpp5p4cs4'Elpp5p4cs4 lpp5p4cs4Epp perfect_unliftr.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
lpp5p4cs4Epp:last ↑[`c[p5]]_p4 cs4 = `c[p]
lpp5cs4'Epp:last `c[p5] cs4' = `c[p]
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size cs1 + (size cs3 + (size cs4).+1).+1
have /IH : last `c[p5] cs4' = `c[p] =>  [//| /(_ Scs4' pp5p4Pcs4') IHc2].
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
lpp5p4cs4Epp:last ↑[`c[p5]]_p4 cs4 = `c[p]
lpp5cs4'Epp:last `c[p5] cs4' = `c[p]
IHc2:size_rpeg `c[p5] p <= size cs4'
                   ?= iff 
                  (cs4' == rpeg `c[p5] p)
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size cs1 + (size cs3 + (size cs4).+1).+1
rewrite size_rpegE rpeg_ppeg // size_ppeg in IHc2.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
lpp5p4cs4Epp:last ↑[`c[p5]]_p4 cs4 = `c[p]
lpp5cs4'Epp:last `c[p5] cs4' = `c[p]
IHc2:(2 ^ n).-1 <= size cs4'
           ?= iff (cs4' == ppeg p5 p)
size_rpeg ↓[c1] `p[p1, p] + 2 ^ n <
size cs1 + (size cs3 + (size cs4).+1).+1
rewrite -[2 ^ n]prednK ?expn_gt0 //.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
lpp5p4cs4Epp:last ↑[`c[p5]]_p4 cs4 = `c[p]
lpp5cs4'Epp:last `c[p5] cs4' = `c[p]
IHc2:(2 ^ n).-1 <= size cs4'
           ?= iff (cs4' == ppeg p5 p)
size_rpeg ↓[c1] `p[p1, p] + (2 ^ n).-1.+1 <
size cs1 + (size cs3 + (size cs4).+1).+1
rewrite !addnS !ltnS.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
lpp5p4cs4Epp:last ↑[`c[p5]]_p4 cs4 = `c[p]
lpp5cs4'Epp:last `c[p5] cs4' = `c[p]
IHc2:(2 ^ n).-1 <= size cs4'
           ?= iff (cs4' == ppeg p5 p)
size_rpeg ↓[c1] `p[p1, p] + (2 ^ n).-1 <=
size cs1 + (size cs3 + size cs4)
apply: leq_trans (leq_addl _ _).
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
lpp5p4cs4Epp:last ↑[`c[p5]]_p4 cs4 = `c[p]
lpp5cs4'Epp:last `c[p5] cs4' = `c[p]
IHc2:(2 ^ n).-1 <= size cs4'
           ?= iff (cs4' == ppeg p5 p)
size_rpeg ↓[c1] `p[p1, p] + (2 ^ n).-1 <=
size cs3 + size cs4
apply: leq_add.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
lpp5p4cs4Epp:last ↑[`c[p5]]_p4 cs4 = `c[p]
lpp5cs4'Epp:last `c[p5] cs4' = `c[p]
IHc2:(2 ^ n).-1 <= size cs4'
           ?= iff (cs4' == ppeg p5 p)
size_rpeg ↓[c1] `p[p1, p] <= size cs3
m:nat
IHm:forall (n : nat) (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) 
  p (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
lpp5p4cs4Epp:last ↑[`c[p5]]_p4 cs4 = `c[p]
lpp5cs4'Epp:last `c[p5] cs4' = `c[p]
IHc2:(2 ^ n).-1 <= size cs4'
           ?= iff (cs4' == ppeg p5 p)
(2 ^ n).-1 <= size cs4
  by apply: leq_trans (size_rpeg_up _ _) _; rewrite IHc1.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
scs4'L:size cs4' <= size cs4
          ?= iff (cs4 ==
                  [seq ↑[i]_
                   (↑[`c[p5]]_p4 ldisk)
                     | i <- cs4'])
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
lpp5p4cs4Epp:last ↑[`c[p5]]_p4 cs4 = `c[p]
lpp5cs4'Epp:last `c[p5] cs4' = `c[p]
IHc2:(2 ^ n).-1 <= size cs4'
           ?= iff (cs4' == ppeg p5 p)
(2 ^ n).-1 <= size cs4
apply: leq_trans scs4'L.
m:nat
IHm:forall (n : nat) (c : configuration 3 n) p (cs : seq (configuration 3 n)),
size cs < m ->
path rmove c cs ->
last c cs = `c[p] -> size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
n:nat
IH:forall (c : configuration 3 n) p
  (cs : seq (configuration 3 n)),
size cs < m.+1 ->
path rmove c cs ->
last c cs = `c[p] ->
size_rpeg c p <= size cs ?= iff (cs == rpeg c p)
c1:configuration 3 n.+1
p:peg 3
cs:seq (configuration 3 n.+1)
Scs:size cs < m.+1
c1nDp:c1 ldisk != p
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1:=c1 ldisk:peg 3
p2:peg 3
p3:=`p[p1, p2]:peg 3
c1':=↓[c1]:configuration 3 n
c2:=↑[`c[p3]]_p2:configuration 3 n.+1
p1Dp2:p1 != p2
p1Rp2:rrel p1 p2
lc1'cs1Epp3:last c1' cs1 = `c[p3]
csE:cs = [seq ↑[i]_p1 | i <- cs1] ++ c2 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1Dp:p1 != p
Scs1:size cs1 < m.+1
Scs2:size cs2 < m.+1
p2Dp:p2 != p
p3Ep:p3 = p
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p4:peg 3
p5:=`p[p2, p4]:peg 3
c2':=↓[c2]:configuration 3 n
p2Dp4:p2 != p4
p2Rp4:rrel p2 p4
c2'cs5Epp5:last c2' cs3 = `c[p5]
cs2E:cs2 =
[seq ↑[i]_p2 | i <- cs3] ++ ↑[`c[p5]]_p4 :: cs4
c2'Pcs3:path (move (rrel (n:=3))) c2' cs3
pp5p4Pcs4:path (move (rrel (n:=3))) ↑[`c[p5]]_p4 cs4
p5Dp3:p5 != p3
cs4':seq (configuration 3 n)
lpp5p4cs4'Elpp5p4cs4:last `c[p5] cs4' =
↓[last ↑[`c[p5]]_p4 cs4]
pp5p4Pcs4':path (move (rrel (n:=3))) `c[p5] cs4'
Scs3:size cs3 < m.+1
Scs4:size cs4 < m.+1
Scs4':size cs4' < m.+1
c2'E:c2' = `c[p3]
IHc1:(2 ^ n).-1 <= size cs3
           ?= iff (cs3 == ppeg p3 p5)
p6Dp:p5 != p
lpp5p4cs4Epp:last ↑[`c[p5]]_p4 cs4 = `c[p]
lpp5cs4'Epp:last `c[p5] cs4' = `c[p]
IHc2:(2 ^ n).-1 <= size cs4'
           ?= iff (cs4' == ppeg p5 p)
(2 ^ n).-1 <= size cs4'
by rewrite IHc2.
Qed.

Lemma gdist_size_rpeg n (c1 : _ _ n) p :  `d[c1, `c[p]]_rmove = size_rpeg c1 p.
n:nat
c1:configuration 3 n
p:peg 3
`d[c1, `c[p]]_rmove = size_rpeg c1 p
Proof.
n:nat
c1:configuration 3 n
p:peg 3
`d[c1, `c[p]]_rmove = size_rpeg c1 p
apply/eqP; rewrite eqn_leq [size_rpeg _ _]size_rpegE.
n:nat
c1:configuration 3 n
p:peg 3
`d[c1, `c[p]]_rmove <= size (rpeg c1 p) <=
`d[c1, `c[p]]_rmove
rewrite gdist_path_le; last 2 first.
n:nat
c1:configuration 3 n
p:peg 3
path rmove c1 (rpeg c1 p)
n:nat
c1:configuration 3 n
p:peg 3
last c1 (rpeg c1 p) = `c[p]
n:nat
c1:configuration 3 n
p:peg 3
true && (size (rpeg c1 p) <= `d[c1, `c[p]]_rmove)
-
n:nat
c1:configuration 3 n
p:peg 3
path rmove c1 (rpeg c1 p)
n:nat
c1:configuration 3 n
p:peg 3
last c1 (rpeg c1 p) = `c[p]
n:nat
c1:configuration 3 n
p:peg 3
true && (size (rpeg c1 p) <= `d[c1, `c[p]]_rmove) by apply: path_rpeg.
n:nat
c1:configuration 3 n
p:peg 3
last c1 (rpeg c1 p) = `c[p]
n:nat
c1:configuration 3 n
p:peg 3
true && (size (rpeg c1 p) <= `d[c1, `c[p]]_rmove)
-
n:nat
c1:configuration 3 n
p:peg 3
last c1 (rpeg c1 p) = `c[p]
n:nat
c1:configuration 3 n
p:peg 3
true && (size (rpeg c1 p) <= `d[c1, `c[p]]_rmove) by rewrite last_rpeg.
n:nat
c1:configuration 3 n
p:peg 3
true && (size (rpeg c1 p) <= `d[c1, `c[p]]_rmove)
have /gpath_connect [p1 p1H] : connect rmove c1 `c[p].
n:nat
c1:configuration 3 n
p:peg 3
c1 `-->*_r `c[p]
n:nat
c1:configuration 3 n
p:peg 3
p1:seq
  (finfun_finType (ordinal_finType n)
     (ordinal_finType 3))
p1H:gpath rmove c1 `c[p] p1
true && (size (rpeg c1 p) <= `d[c1, `c[p]]_rmove)
  by apply: move_connect_rpeg.
n:nat
c1:configuration 3 n
p:peg 3
p1:seq
  (finfun_finType (ordinal_finType n)
     (ordinal_finType 3))
p1H:gpath rmove c1 `c[p] p1
true && (size (rpeg c1 p) <= `d[c1, `c[p]]_rmove)
rewrite -size_rpegE (gpath_dist p1H) /=.
n:nat
c1:configuration 3 n
p:peg 3
p1:seq
  (finfun_finType (ordinal_finType n)
     (ordinal_finType 3))
p1H:gpath rmove c1 `c[p] p1
size_rpeg c1 p <= size p1
apply: (rpeg_min (gpath_path p1H)) => //.
n:nat
c1:configuration 3 n
p:peg 3
p1:seq
  (finfun_finType (ordinal_finType n)
     (ordinal_finType 3))
p1H:gpath rmove c1 `c[p] p1
last c1 p1 = `c[p]
by apply: gpath_last p1H.
Qed.

Lemma gdist_perfect_le n (c : configuration 3 n) p :
   `d[c, `c[p]]_rmove <= (2^ n).-1.
n:nat
c:configuration 3 n
p:peg 3
`d[c, `c[p]]_rmove <= (2 ^ n).-1
Proof.
n:nat
c:configuration 3 n
p:peg 3
`d[c, `c[p]]_rmove <= (2 ^ n).-1 by rewrite gdist_size_rpeg; apply: size_rpeg_up. Qed.

(******************************************************************************)
(* Function that builds a path from a peg to a configuration                  *)
(******************************************************************************)

Definition lpeg n p (c : _ _ n) := rev (belast c (rpeg c p)).

Lemma lpeg_perfect n p : lpeg p `c[p, n]  = [::].
n:nat
p:peg 3
lpeg p `c[p , n] = [::]
Proof.
n:nat
p:peg 3
lpeg p `c[p , n] = [::] by rewrite /lpeg rpeg_perfect. Qed.

Lemma lpeg_nil_inv n c p : 
  lpeg p c = [::] -> c = `c[p] :> configuration _ n.
n:nat
c:configuration 3 n
p:peg 3
lpeg p c = [::] -> c = `c[p]
Proof.
n:nat
c:configuration 3 n
p:peg 3
lpeg p c = [::] -> c = `c[p]
have := @rpeg_nil_inv _ c p.
n:nat
c:configuration 3 n
p:peg 3
(rpeg c p = [::] -> c = `c[p]) ->
lpeg p c = [::] -> c = `c[p]
rewrite /lpeg; case: rpeg => //= a l.
n:nat
c:configuration 3 n
p:peg 3
a:configuration 3 n
l:seq (configuration 3 n)
(a :: l = [::] -> c = `c[p]) ->
rev (c :: belast a l) = [::] -> c = `c[p]
by rewrite rev_cons; case: rev.
Qed.

Lemma path_lpeg n (c : configuration 3 n) p (cs := lpeg p c) :
  path rmove `c[p] cs.
n:nat
c:configuration 3 n
p:peg 3
cs:=lpeg p c:seq (configuration 3 n)
path rmove `c[p] cs
Proof.
n:nat
c:configuration 3 n
p:peg 3
cs:=lpeg p c:seq (configuration 3 n)
path rmove `c[p] cs
have HHs := @rsym 3.
n:nat
c:configuration 3 n
p:peg 3
cs:=lpeg p c:seq (configuration 3 n)
HHs:symmetric (rrel (n:=3))
path rmove `c[p] cs
rewrite {}/cs /lpeg -(last_rpeg c) path_move_rev //.
n:nat
c:configuration 3 n
p:peg 3
HHs:symmetric (rrel (n:=3))
path (move (rrel (n:=3))) c (rpeg c p) 
by apply: path_rpeg.
Qed.

Lemma last_lpeg n (c : configuration 3 n) p (cs := lpeg p c) :
  last `c[p] cs = c.
n:nat
c:configuration 3 n
p:peg 3
cs:=lpeg p c:seq (configuration 3 n)
last `c[p] cs = c
Proof.
n:nat
c:configuration 3 n
p:peg 3
cs:=lpeg p c:seq (configuration 3 n)
last `c[p] cs = c
have HHs := @rsym 3; have := last_rpeg c p.
n:nat
c:configuration 3 n
p:peg 3
cs:=lpeg p c:seq (configuration 3 n)
HHs:symmetric (rrel (n:=3))
(let cs := rpeg c p in last c cs = `c[p]) ->
last `c[p] cs = c
rewrite {}/cs /lpeg; case: rpeg => //= c1 cs.
n:nat
c:configuration 3 n
p:peg 3
HHs:symmetric (rrel (n:=3))
c1:configuration 3 n
cs:seq (configuration 3 n)
last c1 cs = `c[p] ->
last `c[p] (rev (c :: belast c1 cs)) = c
by rewrite rev_cons last_rcons.
Qed.

Lemma size_lpegE n (c : _ _ n) p :
  size_rpeg c p = size (lpeg p c).
n:nat
c:configuration 3 n
p:peg 3
size_rpeg c p = size (lpeg p c)
Proof.
n:nat
c:configuration 3 n
p:peg 3
size_rpeg c p = size (lpeg p c) by rewrite size_rev size_belast size_rpegE. Qed.

Lemma lpeg_min n (c : configuration 3 n) p cs :
  path rmove `c[p] cs -> last `c[p] cs = c ->
  size_rpeg c p <= size cs ?= iff (cs == lpeg p c).
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
path rmove `c[p] cs ->
last `c[p] cs = c ->
size_rpeg c p <= size cs ?= iff (cs == lpeg p c)
Proof.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
path rmove `c[p] cs ->
last `c[p] cs = c ->
size_rpeg c p <= size cs ?= iff (cs == lpeg p c)
(* why this is so complicated???? *)
move=> pPcs lccsEc.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
size_rpeg c p <= size cs ?= iff (cs == lpeg p c)
have HHs := @rsym 3.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
size_rpeg c p <= size cs ?= iff (cs == lpeg p c)
have cPr : path rmove c (rev (belast `c[p] cs)).
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
path rmove c (rev (belast `c[p] cs))
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
size_rpeg c p <= size cs ?= iff (cs == lpeg p c)
  by rewrite -{1}lccsEc path_move_rev.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
size_rpeg c p <= size cs ?= iff (cs == lpeg p c)
have lcrEp : last c (rev (belast `c[p] cs)) = `c[p].
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
last c (rev (belast `c[p] cs)) = `c[p]
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
size_rpeg c p <= size cs ?= iff (cs == lpeg p c)
  rewrite -lccsEc; case: (cs)=> //= c3 cs1.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
c3:configuration 3 n
cs1:seq (configuration 3 n)
last (last c3 cs1) (rev (`c[p] :: belast c3 cs1)) =
`c[p]
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
size_rpeg c p <= size cs ?= iff (cs == lpeg p c)
  by rewrite rev_cons last_rcons.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
size_rpeg c p <= size cs ?= iff (cs == lpeg p c)
have := rpeg_min cPr lcrEp.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
size_rpeg c p <= size (rev (belast `c[p] cs))
              ?= iff (rev (belast `c[p] cs) ==
                      rpeg c p) ->
size_rpeg c p <= size cs ?= iff (cs == lpeg p c)
rewrite /lpeg size_rev size_belast.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
size_rpeg c p <= size cs
              ?= iff (rev (belast `c[p] cs) ==
                      rpeg c p) ->
size_rpeg c p <= size cs
              ?= iff (cs == rev (belast c (rpeg c p)))
set u := rev _ ; set v := rpeg _ _.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
u:=rev (belast `c[p] cs):seq (configuration 3 n)
v:=rpeg c p:seq (configuration 3 n)
size_rpeg c p <= size cs ?= iff (u == v) ->
size_rpeg c p <= size cs
              ?= iff (cs == rev (belast c v))
have -> : (u == v) = (rev (c :: u) == rev (c :: v)).
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
u:=rev (belast `c[p] cs):seq (configuration 3 n)
v:=rpeg c p:seq (configuration 3 n)
(u == v) = (rev (c :: u) == rev (c :: v))
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
u:=rev (belast `c[p] cs):seq (configuration 3 n)
v:=rpeg c p:seq (configuration 3 n)
size_rpeg c p <= size cs
              ?= iff (rev (c :: u) == rev (c :: v)) ->
size_rpeg c p <= size cs
              ?= iff (cs == rev (belast c v))
  rewrite !rev_cons eqseq_rcons eqxx andbT.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
u:=rev (belast `c[p] cs):seq (configuration 3 n)
v:=rpeg c p:seq (configuration 3 n)
(u == v) = (rev u == rev v)
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
u:=rev (belast `c[p] cs):seq (configuration 3 n)
v:=rpeg c p:seq (configuration 3 n)
size_rpeg c p <= size cs
              ?= iff (rev (c :: u) == rev (c :: v)) ->
size_rpeg c p <= size cs
              ?= iff (cs == rev (belast c v))
  apply/eqP/eqP=> [->//|].
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
u:=rev (belast `c[p] cs):seq (configuration 3 n)
v:=rpeg c p:seq (configuration 3 n)
rev u = rev v -> u = v
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
u:=rev (belast `c[p] cs):seq (configuration 3 n)
v:=rpeg c p:seq (configuration 3 n)
size_rpeg c p <= size cs
              ?= iff (rev (c :: u) == rev (c :: v)) ->
size_rpeg c p <= size cs
              ?= iff (cs == rev (belast c v))
  by rewrite -{2}[u]revK => ->; rewrite revK.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
u:=rev (belast `c[p] cs):seq (configuration 3 n)
v:=rpeg c p:seq (configuration 3 n)
size_rpeg c p <= size cs
              ?= iff (rev (c :: u) == rev (c :: v)) ->
size_rpeg c p <= size cs
              ?= iff (cs == rev (belast c v))
rewrite [c :: v]lastI -/v rev_rcons.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
u:=rev (belast `c[p] cs):seq (configuration 3 n)
v:=rpeg c p:seq (configuration 3 n)
size_rpeg c p <= size cs
              ?= iff (rev (c :: u) ==
                      last c v :: rev (belast c v)) ->
size_rpeg c p <= size cs
              ?= iff (cs == rev (belast c v))
rewrite rev_cons revK -{2}lccsEc -lastI eqseq_cons andbC.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
u:=rev (belast `c[p] cs):seq (configuration 3 n)
v:=rpeg c p:seq (configuration 3 n)
size_rpeg c p <= size cs
              ?= iff (cs == rev (belast c v)) &&
                     (`c[p] == last c v) ->
size_rpeg c p <= size cs
              ?= iff (cs == rev (belast c v))
case: eqP=> //; case: eqP => // pDl Hcs; case: pDl.
n:nat
c:configuration 3 n
p:peg 3
cs:seq (configuration 3 n)
pPcs:path rmove `c[p] cs
lccsEc:last `c[p] cs = c
HHs:symmetric (rrel (n:=3))
cPr:path rmove c (rev (belast `c[p] cs))
lcrEp:last c (rev (belast `c[p] cs)) = `c[p]
u:=rev (belast `c[p] cs):seq (configuration 3 n)
v:=rpeg c p:seq (configuration 3 n)
Hcs:cs = rev (belast c v)
`c[p] = last c v
by rewrite last_rpeg.
Qed.

Fixpoint rhanoi3 {n : nat} :=
  if n is n1.+1 return configuration 3 n -> configuration 3 n -> _
  _  then
      fun c1 c2 =>
       let p1 := c1 ldisk in
       let p2 := c2 ldisk in
       let c1' := ↓[c1] in
       let c2' := ↓[c2] in
       if p1 == p2 then [seq ↑[i]_p1 | i <- rhanoi3 c1' c2'] else
       let p := `p[p1, p2] in
       (* one jump *)
       let m1 := size_rpeg c1' p + size_rpeg c2' p in 
       (* two jumps *)
       let m2 := size_rpeg c1' p2 + 2 ^ n1 + size_rpeg c2' p1
       in if m1 <= m2 then 
            [seq ↑[i]_p1 | i <- rpeg c1' p] ++
            [seq ↑[i]_p2 | i <- `c[p] :: lpeg p c2'] 
          else
            [seq ↑[i]_p1 | i <- rpeg c1' p2] ++
            [seq ↑[i]_p | i <- `c[p2] :: ppeg p2 p1] ++
            [seq ↑[i]_p2 | i <- `c[p1] :: lpeg p1 c2']
  else fun _ _ => [::].

Lemma last_rhanoi3 n (c1 c2 : _ _ n) (cs := rhanoi3 c1 c2) :
  last c1 cs = c2.
n:nat
c1, c2:configuration 3 n
cs:=rhanoi3 c1 c2:seq (configuration 3 n)
last c1 cs = c2
Proof.
n:nat
c1, c2:configuration 3 n
cs:=rhanoi3 c1 c2:seq (configuration 3 n)
last c1 cs = c2
have HHr := @rirr 3.
n:nat
c1, c2:configuration 3 n
cs:=rhanoi3 c1 c2:seq (configuration 3 n)
HHr:irreflexive (rrel (n:=3))
last c1 cs = c2
rewrite {}/cs.
n:nat
c1, c2:configuration 3 n
HHr:irreflexive (rrel (n:=3))
last c1 (rhanoi3 c1 c2) = c2
elim: n c1 c2 => /= [c1 c2 |n IH c1 c2]; first by apply/ffunP=> [] [].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
last c1 (rhanoi3 c1 c2) = c2
c1, c2:configuration 3 n.+1
last c1
  (if c1 ldisk == c2 ldisk
   then
    [seq ↑[i]_(c1 ldisk) | i <- rhanoi3 ↓[c1] ↓[c2]]
   else
    if
     size_rpeg ↓[c1] `p[c1 ldisk, c2 ldisk] +
     size_rpeg ↓[c2] `p[c1 ldisk, c2 ldisk] <=
     size_rpeg ↓[c1] (c2 ldisk) + 2 ^ n +
     size_rpeg ↓[c2] (c1 ldisk)
    then
     [seq ↑[i]_(c1 ldisk)
        | i <- rpeg ↓[c1] `p[c1 ldisk, c2 ldisk]] ++
     ↑[`c[`p[c1 ldisk, c2 ldisk]]]_(c2 ldisk)
     :: [seq ↑[i]_(c2 ldisk)
           | i <- lpeg `p[c1 ldisk, c2 ldisk] ↓[c2]]
    else
     [seq ↑[i]_(c1 ldisk)
        | i <- rpeg ↓[c1] (c2 ldisk)] ++
     ↑[`c[c2 ldisk]]_(`p[c1 ldisk, c2 ldisk])
     :: [seq ↑[i]_(`p[c1 ldisk, c2 ldisk])
           | i <- ppeg (c2 ldisk) (c1 ldisk)] ++
        ↑[`c[c1 ldisk]]_(c2 ldisk)
        :: [seq ↑[i]_(c2 ldisk)
              | i <- lpeg (c1 ldisk) ↓[c2]]) = c2
set p1 := _ ldisk; set p2 := _ ldisk.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
last c1 (rhanoi3 c1 c2) = c2
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
last c1
  (if p1 == p2
   then [seq ↑[i]_p1 | i <- rhanoi3 ↓[c1] ↓[c2]]
   else
    if
     size_rpeg ↓[c1] `p[p1, p2] +
     size_rpeg ↓[c2] `p[p1, p2] <=
     size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
    then
     [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p2]] ++
     ↑[`c[`p[p1, p2]]]_p2
     :: [seq ↑[i]_p2 | i <- lpeg `p[p1, p2] ↓[c2]]
    else
     [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
     ↑[`c[p2]]_(`p[p1, p2])
     :: [seq ↑[i]_(`p[p1, p2]) | i <- ppeg p2 p1] ++
        ↑[`c[p1]]_p2
        :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) = c2
set c3 := cliftr _ _; set c4 := cliftr _ _; set c5 := cliftr _ _.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
last c1 (rhanoi3 c1 c2) = c2
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
last c1
  (if p1 == p2
   then [seq ↑[i]_p1 | i <- rhanoi3 ↓[c1] ↓[c2]]
   else
    if
     size_rpeg ↓[c1] `p[p1, p2] +
     size_rpeg ↓[c2] `p[p1, p2] <=
     size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
    then
     [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p2]] ++
     c3 :: [seq ↑[i]_p2 | i <- lpeg `p[p1, p2] ↓[c2]]
    else
     [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
     c4
     :: [seq ↑[i]_(`p[p1, p2]) | i <- ppeg p2 p1] ++
        c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) = c2
set p := `p[_, _].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
last c1 (rhanoi3 c1 c2) = c2
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
last c1
  (if p1 == p2
   then [seq ↑[i]_p1 | i <- rhanoi3 ↓[c1] ↓[c2]]
   else
    if
     size_rpeg ↓[c1] p + size_rpeg ↓[c2] p <=
     size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
    then
     [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] ++
     c3 :: [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
    else
     [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
     c4
     :: [seq ↑[i]_p | i <- ppeg p2 p1] ++
        c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) = c2
case: eqP => [p1Ep2|/eqP p1Dp2].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
last c1 (rhanoi3 c1 c2) = c2
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Ep2:p1 = p2
last c1 [seq ↑[i]_p1 | i <- rhanoi3 ↓[c1] ↓[c2]] = c2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
last c1 (rhanoi3 c1 c2) = c2
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1
  (if
    size_rpeg ↓[c1] p + size_rpeg ↓[c2] p <=
    size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
   then
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] ++
    c3 :: [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
   else
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
    c4
    :: [seq ↑[i]_p | i <- ppeg p2 p1] ++
       c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) = c2
  by rewrite -{1}[c1]cunliftrK last_map IH [c1 _]p1Ep2 cunliftrK.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
last c1 (rhanoi3 c1 c2) = c2
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1
  (if
    size_rpeg ↓[c1] p + size_rpeg ↓[c2] p <=
    size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
   then
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] ++
    c3 :: [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
   else
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
    c4
    :: [seq ↑[i]_p | i <- ppeg p2 p1] ++
       c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) = c2
case: leqP => _; first by rewrite last_cat /= last_map last_lpeg cunliftrK.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
last c1 (rhanoi3 c1 c2) = c2
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1
  ([seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
   c4
   :: [seq ↑[i]_p | i <- ppeg p2 p1] ++
      c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) = c2
by rewrite last_cat /= last_cat /= last_map last_lpeg cunliftrK.
Qed.

Lemma path_rhanoi3 n (c1 c2 : _ _ n) (cs := rhanoi3 c1 c2) :
  path rmove c1 cs.
n:nat
c1, c2:configuration 3 n
cs:=rhanoi3 c1 c2:seq (configuration 3 n)
path rmove c1 cs
Proof.
n:nat
c1, c2:configuration 3 n
cs:=rhanoi3 c1 c2:seq (configuration 3 n)
path rmove c1 cs
have HHr := @rirr 3.
n:nat
c1, c2:configuration 3 n
cs:=rhanoi3 c1 c2:seq (configuration 3 n)
HHr:irreflexive (rrel (n:=3))
path rmove c1 cs
rewrite {}/cs.
n:nat
c1, c2:configuration 3 n
HHr:irreflexive (rrel (n:=3))
path rmove c1 (rhanoi3 c1 c2)
elim: n c1 c2 => //= n IH c1 c2.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
path rmove c1
  (if c1 ldisk == c2 ldisk
   then
    [seq ↑[i]_(c1 ldisk) | i <- rhanoi3 ↓[c1] ↓[c2]]
   else
    if
     size_rpeg ↓[c1] `p[c1 ldisk, c2 ldisk] +
     size_rpeg ↓[c2] `p[c1 ldisk, c2 ldisk] <=
     size_rpeg ↓[c1] (c2 ldisk) + 2 ^ n +
     size_rpeg ↓[c2] (c1 ldisk)
    then
     [seq ↑[i]_(c1 ldisk)
        | i <- rpeg ↓[c1] `p[c1 ldisk, c2 ldisk]] ++
     ↑[`c[`p[c1 ldisk, c2 ldisk]]]_(c2 ldisk)
     :: [seq ↑[i]_(c2 ldisk)
           | i <- lpeg `p[c1 ldisk, c2 ldisk] ↓[c2]]
    else
     [seq ↑[i]_(c1 ldisk)
        | i <- rpeg ↓[c1] (c2 ldisk)] ++
     ↑[`c[c2 ldisk]]_(`p[c1 ldisk, c2 ldisk])
     :: [seq ↑[i]_(`p[c1 ldisk, c2 ldisk])
           | i <- ppeg (c2 ldisk) (c1 ldisk)] ++
        ↑[`c[c1 ldisk]]_(c2 ldisk)
        :: [seq ↑[i]_(c2 ldisk)
              | i <- lpeg (c1 ldisk) ↓[c2]])
set p1 := _ ldisk; set p2 := _ ldisk.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
path rmove c1
  (if p1 == p2
   then [seq ↑[i]_p1 | i <- rhanoi3 ↓[c1] ↓[c2]]
   else
    if
     size_rpeg ↓[c1] `p[p1, p2] +
     size_rpeg ↓[c2] `p[p1, p2] <=
     size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
    then
     [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p2]] ++
     ↑[`c[`p[p1, p2]]]_p2
     :: [seq ↑[i]_p2 | i <- lpeg `p[p1, p2] ↓[c2]]
    else
     [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
     ↑[`c[p2]]_(`p[p1, p2])
     :: [seq ↑[i]_(`p[p1, p2]) | i <- ppeg p2 p1] ++
        ↑[`c[p1]]_p2
        :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
set c3 := cliftr _ _; set c4 := cliftr _ _; set c5 := cliftr _ _.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
path rmove c1
  (if p1 == p2
   then [seq ↑[i]_p1 | i <- rhanoi3 ↓[c1] ↓[c2]]
   else
    if
     size_rpeg ↓[c1] `p[p1, p2] +
     size_rpeg ↓[c2] `p[p1, p2] <=
     size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
    then
     [seq ↑[i]_p1 | i <- rpeg ↓[c1] `p[p1, p2]] ++
     c3 :: [seq ↑[i]_p2 | i <- lpeg `p[p1, p2] ↓[c2]]
    else
     [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
     c4
     :: [seq ↑[i]_(`p[p1, p2]) | i <- ppeg p2 p1] ++
        c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
set p := `p[_, _].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
path rmove c1
  (if p1 == p2
   then [seq ↑[i]_p1 | i <- rhanoi3 ↓[c1] ↓[c2]]
   else
    if
     size_rpeg ↓[c1] p + size_rpeg ↓[c2] p <=
     size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
    then
     [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] ++
     c3 :: [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
    else
     [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
     c4
     :: [seq ↑[i]_p | i <- ppeg p2 p1] ++
        c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
case: eqP => [p1Ep2|/eqP p1Dp2].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Ep2:p1 = p2
path rmove c1 [seq ↑[i]_p1 | i <- rhanoi3 ↓[c1] ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1
  (if
    size_rpeg ↓[c1] p + size_rpeg ↓[c2] p <=
    size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
   then
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] ++
    c3 :: [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
   else
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
    c4
    :: [seq ↑[i]_p | i <- ppeg p2 p1] ++
       c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  by rewrite -{1}[c1]cunliftrK path_liftr.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1
  (if
    size_rpeg ↓[c1] p + size_rpeg ↓[c2] p <=
    size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
   then
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] ++
    c3 :: [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
   else
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
    c4
    :: [seq ↑[i]_p | i <- ppeg p2 p1] ++
       c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
case: leqP => _; rewrite !cat_path /=; apply/and3P; split.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] `-->_r c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
-
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] `-->_r c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) by rewrite -{1}[c1]cunliftrK path_liftr // path_rpeg.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] `-->_r c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
-
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] `-->_r c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) rewrite -{1}[c1]cunliftrK last_map last_rpeg.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
↑[`c[p]]_(c1 ldisk) `-->_r c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  apply/moveP; exists ldisk.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
[/\ rrel (↑[`c[p]]_(c1 ldisk) ldisk) (c3 ldisk),
    forall d2 : ordinal_finType n.+1,
    ldisk != d2 -> ↑[`c[p]]_(c1 ldisk) d2 = c3 d2,
    on_top ldisk ↑[`c[p]]_(c1 ldisk)
  & on_top ldisk c3]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  split => // [|d dmDd||].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
rrel (↑[`c[p]]_(c1 ldisk) ldisk) (c3 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d:ordinal_finType n.+1
dmDd:ldisk != d
↑[`c[p]]_(c1 ldisk) d = c3 d
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p]]_(c1 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  -
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
rrel (↑[`c[p]]_(c1 ldisk) ldisk) (c3 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d:ordinal_finType n.+1
dmDd:ldisk != d
↑[`c[p]]_(c1 ldisk) d = c3 d
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p]]_(c1 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) by rewrite !cliftr_ldisk.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d:ordinal_finType n.+1
dmDd:ldisk != d
↑[`c[p]]_(c1 ldisk) d = c3 d
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p]]_(c1 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  -
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d:ordinal_finType n.+1
dmDd:ldisk != d
↑[`c[p]]_(c1 ldisk) d = c3 d
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p]]_(c1 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) rewrite !ffunE; case: tsplitP => [j|j jE]; first by rewrite !ffunE.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d:ordinal_finType n.+1
dmDd:ldisk != d
j:'I_1
jE:d = j + n
`c[c1 ldisk] j = `c[p2] j
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p]]_(c1 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
    by case/eqP: dmDd; apply/val_eqP; rewrite /= jE; case: (j) => [] [].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p]]_(c1 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  -
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p]]_(c1 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) apply/on_topP=> d; rewrite !cliftr_ldisk !ffunE.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d:ordinal_finType n.+1
c1 ldisk =
match tsplit d with
| inl j => `c[c1 ldisk] j
| inr j => `c[p] j
end -> ldisk <= d
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
    case: tsplitP => [j _ /eqP|j jE].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d:ordinal_finType n.+1
j:'I_n
c1 ldisk == `c[p] j -> ldisk <= d
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d:ordinal_finType n.+1
j:'I_1
jE:d = j + n
c1 ldisk = `c[c1 ldisk] j -> ldisk <= d
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
      by rewrite !ffunE -/p1 eq_sym (negPf (opegDl _ _)).
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d:ordinal_finType n.+1
j:'I_1
jE:d = j + n
c1 ldisk = `c[c1 ldisk] j -> ldisk <= d
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
    by rewrite /= !ffunE jE /= leq_addl.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c3
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) 
  apply/on_topP=> d; rewrite /= !cliftr_ldisk !ffunE.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d:ordinal_finType n.+1
p2 =
match tsplit d with
| inl j => `c[p2] j
| inr j => `c[`p[p1, p2]] j
end -> n <= d
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  case: tsplitP => [j _ /eqP|j ->]; last by case: j => [] [].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d:ordinal_finType n.+1
j:'I_n
p2 == `c[`p[p1, p2]] j -> n <= d
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  by rewrite !ffunE eq_sym (negPf (opegDr _ _)).
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])  
-
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c3 [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) by rewrite -[c3]cunliftrK !cliftr_ldisk /= path_liftr // cliftrK path_lpeg.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
-
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) by rewrite -{1}[c1]cunliftrK path_liftr // path_rpeg.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
-
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c1 [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) rewrite -{1}[c1]cunliftrK last_map last_rpeg.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
↑[`c[p2]]_(c1 ldisk) `-->_r c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  apply/moveP; exists ldisk; split => // [|d2||]; 
         rewrite ?cliftr_ldisk ?ffunE //=.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
rrel (c1 ldisk) `p[p1, p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
ldisk != d2 ->
match tsplit d2 with
| inl j => `c[c1 ldisk] j
| inr j => `c[p2] j
end =
match tsplit d2 with
| inl j => `c[`p[p1, p2]] j
| inr j => `c[p2] j
end
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p2]]_(c1 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  -
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
rrel (c1 ldisk) `p[p1, p2]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
ldisk != d2 ->
match tsplit d2 with
| inl j => `c[c1 ldisk] j
| inr j => `c[p2] j
end =
match tsplit d2 with
| inl j => `c[`p[p1, p2]] j
| inr j => `c[p2] j
end
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p2]]_(c1 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) by rewrite /rrel /= eq_sym opegDl.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
ldisk != d2 ->
match tsplit d2 with
| inl j => `c[c1 ldisk] j
| inr j => `c[p2] j
end =
match tsplit d2 with
| inl j => `c[`p[p1, p2]] j
| inr j => `c[p2] j
end
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p2]]_(c1 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  -
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
ldisk != d2 ->
match tsplit d2 with
| inl j => `c[c1 ldisk] j
| inr j => `c[p2] j
end =
match tsplit d2 with
| inl j => `c[`p[p1, p2]] j
| inr j => `c[p2] j
end
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p2]]_(c1 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) case: tsplitP => //= j d2E /eqP[].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
j:'I_1
d2E:d2 = j + n
ldisk = d2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p2]]_(c1 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
    by apply/val_eqP; rewrite /= d2E; case: (j) => [] [].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p2]]_(c1 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  -
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p2]]_(c1 ldisk)
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) apply/on_topP=> d2.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
↑[`c[p2]]_(c1 ldisk) ldisk = ↑[`c[p2]]_(c1 ldisk) d2 ->
ldisk <= d2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
    rewrite cliftr_ldisk /= !ffunE.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
c1 ldisk =
match tsplit d2 with
| inl j => `c[c1 ldisk] j
| inr j => `c[p2] j
end -> n <= d2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
    case: tsplitP => [k _ /eqP | [[]] // j -> //].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
k:'I_n
c1 ldisk == `c[p2] k -> n <= d2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
    by rewrite ?ffunE (negPf p1Dp2).
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c4
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  apply/on_topP=> d2.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
c4 ldisk = c4 d2 -> ldisk <= d2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  rewrite cliftr_ldisk /= !ffunE.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
`p[p1, p2] =
match tsplit d2 with
| inl j => `c[`p[p1, p2]] j
| inr j => `c[p2] j
end -> n <= d2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  case: tsplitP => [k _ /eqP | [[]] // j -> //].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
k:'I_n
`p[p1, p2] == `c[p2] k -> n <= d2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
  by rewrite !ffunE (negPf (opegDr _ _)).
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4
  ([seq ↑[i]_p | i <- ppeg p2 p1] ++
   c5 :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]])
rewrite cat_path /=; apply/and3P; split => //.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4 [seq ↑[i]_p | i <- ppeg p2 p1]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c4 [seq ↑[i]_p | i <- ppeg p2 p1] `-->_r c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
-
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c4 [seq ↑[i]_p | i <- ppeg p2 p1]
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c4 [seq ↑[i]_p | i <- ppeg p2 p1] `-->_r c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]] rewrite -[c4]cunliftrK cliftr_ldisk -/p path_liftr // cliftrK // path_ppeg //.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
p2 != p1
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c4 [seq ↑[i]_p | i <- ppeg p2 p1] `-->_r c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
  by rewrite eq_sym.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c4 [seq ↑[i]_p | i <- ppeg p2 p1] `-->_r c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
-
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
last c4 [seq ↑[i]_p | i <- ppeg p2 p1] `-->_r c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]] rewrite -[c4]cunliftrK cliftr_ldisk -/p last_map cliftrK // last_ppeg //.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
↑[`c[p1]]_p `-->_r c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
  apply/moveP; exists ldisk; split => // [|d2||]; 
         rewrite ?cliftr_ldisk ?ffunE /=.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
rrel p p2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
ldisk != d2 ->
match tsplit d2 with
| inl j => `c[p] j
| inr j => `c[p1] j
end =
match tsplit d2 with
| inl j => `c[p2] j
| inr j => `c[p1] j
end
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p1]]_p
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
  -
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
rrel p p2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
ldisk != d2 ->
match tsplit d2 with
| inl j => `c[p] j
| inr j => `c[p1] j
end =
match tsplit d2 with
| inl j => `c[p2] j
| inr j => `c[p1] j
end
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p1]]_p
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]] by rewrite /rrel /= opegDr.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
ldisk != d2 ->
match tsplit d2 with
| inl j => `c[p] j
| inr j => `c[p1] j
end =
match tsplit d2 with
| inl j => `c[p2] j
| inr j => `c[p1] j
end
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p1]]_p
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
  -
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
ldisk != d2 ->
match tsplit d2 with
| inl j => `c[p] j
| inr j => `c[p1] j
end =
match tsplit d2 with
| inl j => `c[p2] j
| inr j => `c[p1] j
end
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p1]]_p
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]] case: tsplitP => [j | j d2E /eqP[]] //=.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
j:'I_1
d2E:d2 = j + n
ldisk = d2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p1]]_p
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
    by apply/val_eqP => /=; rewrite d2E; case: (j) => [] [].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p1]]_p
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
  -
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk ↑[`c[p1]]_p
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]] apply/on_topP=> d2.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
↑[`c[p1]]_p ldisk = ↑[`c[p1]]_p d2 -> ldisk <= d2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
    rewrite cliftr_ldisk /= !ffunE.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
p =
match tsplit d2 with
| inl j => `c[p] j
| inr j => `c[p1] j
end -> n <= d2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
    case: tsplitP => [k _ /eqP | [[]] // j -> //].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
k:'I_n
p == `c[p1] k -> n <= d2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
    by rewrite !ffunE (negPf (opegDl _ _)).
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
on_top ldisk c5
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
  apply/on_topP=> d2.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
c5 ldisk = c5 d2 -> ldisk <= d2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
  rewrite cliftr_ldisk /= !ffunE.
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
p2 =
match tsplit d2 with
| inl j => `c[p2] j
| inr j => `c[p1] j
end -> n <= d2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
  case: tsplitP => [k _ /eqP | [[]] // j -> //].
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
d2:ordinal_finType n.+1
k:'I_n
p2 == `c[p1] k -> n <= d2
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
  by rewrite !ffunE eq_sym (negPf p1Dp2).
HHr:irreflexive (rrel (n:=3))
n:nat
IH:forall c1 c2 : configuration 3 n,
path rmove c1 (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
c3:=↑[`c[`p[p1, p2]]]_p2:configuration 3 n.+1
c4:=↑[`c[p2]]_(`p[p1, p2]):configuration 3 n.+1
c5:=↑[`c[p1]]_p2:configuration 3 n.+1
p:=`p[p1, p2]:peg 3
p1Dp2:p1 != p2
path rmove c5 [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]
by rewrite path_liftr // path_lpeg.
Qed.

Lemma rhanoi3_min n (c1 c2 : configuration 3 n) cs :
  path rmove c1 cs -> last c1 cs = c2 ->
  size (rhanoi3 c1 c2) <= size cs.
n:nat
c1, c2:configuration 3 n
cs:seq (configuration 3 n)
path rmove c1 cs ->
last c1 cs = c2 -> size (rhanoi3 c1 c2) <= size cs
Proof.
n:nat
c1, c2:configuration 3 n
cs:seq (configuration 3 n)
path rmove c1 cs ->
last c1 cs = c2 -> size (rhanoi3 c1 c2) <= size cs
have HHr := @rirr 3.
n:nat
c1, c2:configuration 3 n
cs:seq (configuration 3 n)
HHr:irreflexive (rrel (n:=3))
path rmove c1 cs ->
last c1 cs = c2 -> size (rhanoi3 c1 c2) <= size cs
have HHs := @rsym 3.
n:nat
c1, c2:configuration 3 n
cs:seq (configuration 3 n)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
path rmove c1 cs ->
last c1 cs = c2 -> size (rhanoi3 c1 c2) <= size cs
elim: n c1 c2 cs => [p c1 [|]//|n IH c1 c2 cs /=].
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
path rmove c1 cs ->
last c1 cs = c2 ->
size
  (if c1 ldisk == c2 ldisk
   then
    [seq ↑[i]_(c1 ldisk) | i <- rhanoi3 ↓[c1] ↓[c2]]
   else
    if
     size_rpeg ↓[c1] `p[c1 ldisk, c2 ldisk] +
     size_rpeg ↓[c2] `p[c1 ldisk, c2 ldisk] <=
     size_rpeg ↓[c1] (c2 ldisk) + 2 ^ n +
     size_rpeg ↓[c2] (c1 ldisk)
    then
     [seq ↑[i]_(c1 ldisk)
        | i <- rpeg ↓[c1] `p[c1 ldisk, c2 ldisk]] ++
     ↑[`c[`p[c1 ldisk, c2 ldisk]]]_(c2 ldisk)
     :: [seq ↑[i]_(c2 ldisk)
           | i <- lpeg `p[c1 ldisk, c2 ldisk] ↓[c2]]
    else
     [seq ↑[i]_(c1 ldisk)
        | i <- rpeg ↓[c1] (c2 ldisk)] ++
     ↑[`c[c2 ldisk]]_(`p[c1 ldisk, c2 ldisk])
     :: [seq ↑[i]_(`p[c1 ldisk, c2 ldisk])
           | i <- ppeg (c2 ldisk) (c1 ldisk)] ++
        ↑[`c[c1 ldisk]]_(c2 ldisk)
        :: [seq ↑[i]_(c2 ldisk)
              | i <- lpeg (c1 ldisk) ↓[c2]]) <=
size cs
set p := `p[_, _]; set p1 := _ ldisk; set p2 := _ ldisk.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
path rmove c1 cs ->
last c1 cs = c2 ->
size
  (if p1 == p2
   then [seq ↑[i]_p1 | i <- rhanoi3 ↓[c1] ↓[c2]]
   else
    if
     size_rpeg ↓[c1] p + size_rpeg ↓[c2] p <=
     size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
    then
     [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] ++
     ↑[`c[p]]_p2 :: [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
    else
     [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
     ↑[`c[p2]]_p
     :: [seq ↑[i]_p | i <- ppeg p2 p1] ++
        ↑[`c[p1]]_p2
        :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) <=
size cs
case: eqP => [p1Ep2 c1Pcs lc1csEc2|/eqP p1Dp2 c1Pcs lc1csEc2].
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Ep2:p1 = p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
size [seq ↑[i]_p1 | i <- rhanoi3 ↓[c1] ↓[c2]] <=
size cs
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
size
  (if
    size_rpeg ↓[c1] p + size_rpeg ↓[c2] p <=
    size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
   then
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] ++
    ↑[`c[p]]_p2 :: [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
   else
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
    ↑[`c[p2]]_p
    :: [seq ↑[i]_p | i <- ppeg p2 p1] ++
       ↑[`c[p1]]_p2
       :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) <=
size cs
  have lcsmEc1m : last c1 cs ldisk = p1 by rewrite lc1csEc2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Ep2:p1 = p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
lcsmEc1m:last c1 cs ldisk = p1
size [seq ↑[i]_p1 | i <- rhanoi3 ↓[c1] ↓[c2]] <=
size cs
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
size
  (if
    size_rpeg ↓[c1] p + size_rpeg ↓[c2] p <=
    size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
   then
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] ++
    ↑[`c[p]]_p2 :: [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
   else
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
    ↑[`c[p2]]_p
    :: [seq ↑[i]_p | i <- ppeg p2 p1] ++
       ↑[`c[p1]]_p2
       :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) <=
size cs
  have [cs1 [c1Pcs1 lc1csElcs1 /leq_of_leqif/(leq_trans _)->//]] := 
      pathS_restrict (@rirr 3) c1Pcs.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Ep2:p1 = p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
lcsmEc1m:last c1 cs ldisk = p1
cs1:seq (configuration 3 n)
c1Pcs1:path (move (rrel (n:=3))) ↓[c1] cs1
lc1csElcs1:last ↓[c1] cs1 = ↓[last c1 cs]
size [seq ↑[i]_p1 | i <- rhanoi3 ↓[c1] ↓[c2]] <=
size cs1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
size
  (if
    size_rpeg ↓[c1] p + size_rpeg ↓[c2] p <=
    size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
   then
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] ++
    ↑[`c[p]]_p2 :: [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
   else
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
    ↑[`c[p2]]_p
    :: [seq ↑[i]_p | i <- ppeg p2 p1] ++
       ↑[`c[p1]]_p2
       :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) <=
size cs
  by rewrite size_map IH // lc1csElcs1 lc1csEc2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
size
  (if
    size_rpeg ↓[c1] p + size_rpeg ↓[c2] p <=
    size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1
   then
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] ++
    ↑[`c[p]]_p2 :: [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
   else
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
    ↑[`c[p2]]_p
    :: [seq ↑[i]_p | i <- ppeg p2 p1] ++
       ↑[`c[p1]]_p2
       :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) <=
size cs
set u := _ + _; set v := _ + _.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
size
  (if u <= v
   then
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] ++
    ↑[`c[p]]_p2 :: [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
   else
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
    ↑[`c[p2]]_p
    :: [seq ↑[i]_p | i <- ppeg p2 p1] ++
       ↑[`c[p1]]_p2
       :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) <=
size cs
suff : minn u v < size cs.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
minn u v < size cs ->
size
  (if u <= v
   then
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p] ++
    ↑[`c[p]]_p2 :: [seq ↑[i]_p2 | i <- lpeg p ↓[c2]]
   else
    [seq ↑[i]_p1 | i <- rpeg ↓[c1] p2] ++
    ↑[`c[p2]]_p
    :: [seq ↑[i]_p | i <- ppeg p2 p1] ++
       ↑[`c[p1]]_p2
       :: [seq ↑[i]_p2 | i <- lpeg p1 ↓[c2]]) <=
size cs
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
minn u v < size cs
  rewrite minnC /minn; case: (leqP u) => H1;
      rewrite !size_cat /= ?size_cat /= !addnS ?ltnS
              !size_map -size_rpegE -size_lpegE -/u => Lscs.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
H1:u <= v
Lscs:u < size cs
u < size cs
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
H1:v < u
Lscs:v < size cs
(size_rpeg ↓[c1] p2 +
 (size (ppeg p2 p1) + size_rpeg ↓[c2] p1)).+1 <
size cs
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
minn u v < size cs
    by apply: leq_trans Lscs.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
H1:v < u
Lscs:v < size cs
(size_rpeg ↓[c1] p2 +
 (size (ppeg p2 p1) + size_rpeg ↓[c2] p1)).+1 <
size cs
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
minn u v < size cs
  apply: leq_ltn_trans _ Lscs.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
H1:v < u
size_rpeg ↓[c1] p2 +
(size (ppeg p2 p1) + size_rpeg ↓[c2] p1) < v
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
minn u v < size cs
  rewrite -addnS -addSn !addnA !leq_add //.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
H1:v < u
size (ppeg p2 p1) < 2 ^ n
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
minn u v < size cs
  by rewrite -[2 ^ _]prednK ?expn_gt0 // ltnS size_ppeg.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
minn u v < size cs
pose f (c : configuration 3 n.+1) := c ldisk.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
cs:seq (configuration 3 n.+1)
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
minn u v < size cs
have [m Lcs] := ubnP (size cs); elim: m => // m IH1 in cs Lcs c1Pcs lc1csEc2 *.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
Lcs:size cs < m.+1
c1Pcs:path rmove c1 cs
lc1csEc2:last c1 cs = c2
minn u v < size cs
case: path3SP c1Pcs => // [c1' cs' /= csE c1'Pcs' _|
                           cs1 cs2 p1' p3 p4 c1' c3 p1Dp3 p1Rp3 lc1'cs1Epp4 csE
                          c1'Pcs1 c3Pcs2 _].
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
Lcs:size cs < m.+1
lc1csEc2:last c1 cs = c2
c1':=↓[c1]:configuration 3 n
cs':=[seq ↓[i] | i <- cs]:seq (configuration 3 n)
csE:cs = [seq ↑[i]_(c1 ldisk) | i <- cs']
c1'Pcs':path (move (rrel (n:=3))) c1' cs'
minn u v < size cs
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
Lcs:size cs < m.+1
lc1csEc2:last c1 cs = c2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
minn u v < size cs
  case/eqP: p1Dp2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m -> path rmove c1 cs -> last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
Lcs:size cs < m.+1
lc1csEc2:last c1 cs = c2
c1':=↓[c1]:configuration 3 n
cs':=[seq ↓[i] | i <- cs]:seq (configuration 3 n)
csE:cs = [seq ↑[i]_(c1 ldisk) | i <- cs']
c1'Pcs':path (move (rrel (n:=3))) c1' cs'
p1 = p2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
Lcs:size cs < m.+1
lc1csEc2:last c1 cs = c2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
minn u v < size cs
  have := congr1 f lc1csEc2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m -> path rmove c1 cs -> last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
Lcs:size cs < m.+1
lc1csEc2:last c1 cs = c2
c1':=↓[c1]:configuration 3 n
cs':=[seq ↓[i] | i <- cs]:seq (configuration 3 n)
csE:cs = [seq ↑[i]_(c1 ldisk) | i <- cs']
c1'Pcs':path (move (rrel (n:=3))) c1' cs'
f (last c1 cs) = f c2 -> p1 = p2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
Lcs:size cs < m.+1
lc1csEc2:last c1 cs = c2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
minn u v < size cs
  rewrite /f  csE -{1}[c1](cunliftrK) last_map.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m -> path rmove c1 cs -> last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
Lcs:size cs < m.+1
lc1csEc2:last c1 cs = c2
c1':=↓[c1]:configuration 3 n
cs':=[seq ↓[i] | i <- cs]:seq (configuration 3 n)
csE:cs = [seq ↑[i]_(c1 ldisk) | i <- cs']
c1'Pcs':path (move (rrel (n:=3))) c1' cs'
↑[last ↓[c1] cs']_(c1 ldisk) ldisk = c2 ldisk ->
p1 = p2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
Lcs:size cs < m.+1
lc1csEc2:last c1 cs = c2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
minn u v < size cs
  by rewrite !cliftr_ldisk.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
Lcs:size cs < m.+1
lc1csEc2:last c1 cs = c2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
minn u v < size cs
have p1'Ep1 : p1' = p1 by [].
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
Lcs:size cs < m.+1
lc1csEc2:last c1 cs = c2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
minn u v < size cs
move: Lcs; rewrite csE size_cat size_map /= addnS ltnS => Lcs.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
lc1csEc2:last c1 cs = c2
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
minn u v < (size cs1 + size cs2).+1
move: lc1csEc2; rewrite csE last_cat /= => lc3cs2Ec2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
minn u v < (size cs1 + size cs2).+1
have [/eqP p3Ep2 | p3Dp2] := boolP (p3 == p2).
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
minn u v < (size cs1 + size cs2).+1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
minn u v < (size cs1 + size cs2).+1
  have p4Ep : p4 = p.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p4 = p
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p4Ep:p4 = p
minn u v < (size cs1 + size cs2).+1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
minn u v < (size cs1 + size cs2).+1
    apply/eqP; rewrite opeg3E // eq_sym opeg3E // negb_and !negbK eqxx /=.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p3 != p
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p4Ep:p4 = p
minn u v < (size cs1 + size cs2).+1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
minn u v < (size cs1 + size cs2).+1
    by rewrite eq_sym opeg3E // p1Dp3 eq_sym p3Ep2 eqxx.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p4Ep:p4 = p
minn u v < (size cs1 + size cs2).+1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
minn u v < (size cs1 + size cs2).+1
  have [cs2' [pp4Pcs2' pp4cs2'E scs2'Lscs2]]:= pathS_restrict HHr c3Pcs2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p4Ep:p4 = p
cs2':seq (configuration 3 n)
pp4Pcs2':path (move (rrel (n:=3))) ↓[c3] cs2'
pp4cs2'E:last ↓[c3] cs2' = ↓[last c3 cs2]
scs2'Lscs2:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c3 ldisk)
                     | i <- cs2'])
minn u v < (size cs1 + size cs2).+1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
minn u v < (size cs1 + size cs2).+1
  rewrite cliftrK in pp4Pcs2' pp4cs2'E.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p4Ep:p4 = p
cs2':seq (configuration 3 n)
scs2'Lscs2:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c3 ldisk)
                     | i <- cs2'])
pp4Pcs2':path (move (rrel (n:=3))) `c[p4] cs2'
pp4cs2'E:last `c[p4] cs2' = ↓[last c3 cs2]
minn u v < (size cs1 + size cs2).+1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
minn u v < (size cs1 + size cs2).+1
  apply: leq_trans (leq_add (leqnn _) scs2'Lscs2).
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p4Ep:p4 = p
cs2':seq (configuration 3 n)
scs2'Lscs2:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c3 ldisk)
                     | i <- cs2'])
pp4Pcs2':path (move (rrel (n:=3))) `c[p4] cs2'
pp4cs2'E:last `c[p4] cs2' = ↓[last c3 cs2]
minn u v <= size cs1 + size cs2'
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
minn u v < (size cs1 + size cs2).+1
  apply: leq_trans (geq_minl _ _) _.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p4Ep:p4 = p
cs2':seq (configuration 3 n)
scs2'Lscs2:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c3 ldisk)
                     | i <- cs2'])
pp4Pcs2':path (move (rrel (n:=3))) `c[p4] cs2'
pp4cs2'E:last `c[p4] cs2' = ↓[last c3 cs2]
u <= size cs1 + size cs2'
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
minn u v < (size cs1 + size cs2).+1
  apply: leq_add.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p4Ep:p4 = p
cs2':seq (configuration 3 n)
scs2'Lscs2:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c3 ldisk)
                     | i <- cs2'])
pp4Pcs2':path (move (rrel (n:=3))) `c[p4] cs2'
pp4cs2'E:last `c[p4] cs2' = ↓[last c3 cs2]
size_rpeg ↓[c1] p <= size cs1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p4Ep:p4 = p
cs2':seq (configuration 3 n)
scs2'Lscs2:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c3 ldisk)
                     | i <- cs2'])
pp4Pcs2':path (move (rrel (n:=3))) `c[p4] cs2'
pp4cs2'E:last `c[p4] cs2' = ↓[last c3 cs2]
size_rpeg ↓[c2] p <= size cs2'
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
minn u v < (size cs1 + size cs2).+1
    apply: leq_of_leqif (rpeg_min _ _ ) => //.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p4Ep:p4 = p
cs2':seq (configuration 3 n)
scs2'Lscs2:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c3 ldisk)
                     | i <- cs2'])
pp4Pcs2':path (move (rrel (n:=3))) `c[p4] cs2'
pp4cs2'E:last `c[p4] cs2' = ↓[last c3 cs2]
last ↓[c1] cs1 = `c[p]
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p4Ep:p4 = p
cs2':seq (configuration 3 n)
scs2'Lscs2:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c3 ldisk)
                     | i <- cs2'])
pp4Pcs2':path (move (rrel (n:=3))) `c[p4] cs2'
pp4cs2'E:last `c[p4] cs2' = ↓[last c3 cs2]
size_rpeg ↓[c2] p <= size cs2'
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
minn u v < (size cs1 + size cs2).+1
      by rewrite lc1'cs1Epp4; congr `c[`p[_, _]].
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p4Ep:p4 = p
cs2':seq (configuration 3 n)
scs2'Lscs2:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c3 ldisk)
                     | i <- cs2'])
pp4Pcs2':path (move (rrel (n:=3))) `c[p4] cs2'
pp4cs2'E:last `c[p4] cs2' = ↓[last c3 cs2]
size_rpeg ↓[c2] p <= size cs2'
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
minn u v < (size cs1 + size cs2).+1
  apply: leq_of_leqif (lpeg_min _ _ ); rewrite -p4Ep //.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Ep2:p3 = p2
p4Ep:p4 = p
cs2':seq (configuration 3 n)
scs2'Lscs2:size cs2' <= size cs2
          ?= iff (cs2 ==
                  [seq ↑[i]_
                   (c3 ldisk)
                     | i <- cs2'])
pp4Pcs2':path (move (rrel (n:=3))) `c[p4] cs2'
pp4cs2'E:last `c[p4] cs2' = ↓[last c3 cs2]
last `c[p4] cs2' = ↓[c2]
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
minn u v < (size cs1 + size cs2).+1
  by rewrite pp4cs2'E lc3cs2Ec2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
minn u v < (size cs1 + size cs2).+1
have p3Ep : p3 = p.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3 = p
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
minn u v < (size cs1 + size cs2).+1
  by apply/eqP; rewrite eq_sym opeg3E // -/p1 -p1'Ep1 p1Dp3 eq_sym.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
minn u v < (size cs1 + size cs2).+1
have p4Ep2 : p4 = p2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4 = p2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
minn u v < (size cs1 + size cs2).+1
  apply/eqP.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4 == p2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
minn u v < (size cs1 + size cs2).+1
  rewrite /p4 p3Ep /p -/p1 -/p2 p1'Ep1 !opeg3E //.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
(p1 != p2) && ~~ ((p1 != p2) && (p2 != p2))
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p1 != `p[p1, p2]
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
minn u v < (size cs1 + size cs2).+1
    by rewrite p1Dp2 negbK eqxx.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p1 != `p[p1, p2]
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
minn u v < (size cs1 + size cs2).+1
  by rewrite eq_sym opeg3E // eqxx eq_sym.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
c3Pcs2:path (move (rrel (n:=3))) c3 cs2
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
minn u v < (size cs1 + size cs2).+1
case: path3SP c3Pcs2=> // [c3' cs2' p5 cs2E c3'Pcs2' _|
                          cs3 cs4 p5 p6 p7 c3' c4 p5Dp6 p5p6
                          c3'cs3Epp7 cs2E c3'Pcs3 c4Pcs4 _].
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
c3':=↓[c3]:configuration 3 n
cs2':=[seq ↓[i] | i <- cs2]:seq (configuration 3 n)
p5:=c3 ldisk:peg 3
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs2']
c3'Pcs2':path (move (rrel (n:=3))) c3' cs2'
minn u v < (size cs1 + size cs2).+1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
minn u v < (size cs1 + size cs2).+1
  have  := congr1 f lc3cs2Ec2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
c3':=↓[c3]:configuration 3 n
cs2':=[seq ↓[i] | i <- cs2]:seq (configuration 3 n)
p5:=c3 ldisk:peg 3
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs2']
c3'Pcs2':path (move (rrel (n:=3))) c3' cs2'
f (last c3 cs2) = f c2 ->
minn u v < (size cs1 + size cs2).+1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
minn u v < (size cs1 + size cs2).+1
  rewrite cs2E /f -[c3]cunliftrK last_map !cliftr_ldisk -/p2 /= 
        => p3Ep2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
c3':=↓[c3]:configuration 3 n
cs2':=[seq ↓[i] | i <- cs2]:seq (configuration 3 n)
p5:=c3 ldisk:peg 3
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs2']
c3'Pcs2':path (move (rrel (n:=3))) c3' cs2'
p3Ep2:p3 = p2
minn u v <
(size cs1 + size [seq ↑[i]_p5 | i <- cs2']).+1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
minn u v < (size cs1 + size cs2).+1
  by case/eqP: p3Dp2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
minn u v < (size cs1 + size cs2).+1
have p5Ep: p5 = p by rewrite /p5 /c3 !cliftr_ldisk.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
Lcs:size cs1 + size cs2 < m
lc3cs2Ec2:last c3 cs2 = c2
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
minn u v < (size cs1 + size cs2).+1
move: lc3cs2Ec2 Lcs; rewrite cs2E last_cat /= size_cat /= size_map => 
           lc6cs4Ec2 Lcs.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
minn u v < (size cs1 + (size cs3 + (size cs4).+1)).+1
rewrite -addSnnS ltnS.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
have [/eqP p6Ep1|p6Dp1] := boolP (p6 == p1).
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
  have p7Ep2 : p7 = p2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7 = p2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
    apply/eqP.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7 == p2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
    by rewrite opeg3E // p5Ep p6Ep1 opeg3E // p1Dp2 eqxx.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
  apply: leq_trans (_ : size (map (cliftr p1) cs1 ++ cs4) <= _).
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
minn u v <= size ([seq ↑[i]_p1 | i <- cs1] ++ cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
size ([seq ↑[i]_p1 | i <- cs1] ++ cs4) <=
size cs1 + ((size cs3).+1 + size cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
    apply/ltnW/IH1.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
size ([seq ↑[i]_p1 | i <- cs1] ++ cs4) < m
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
path rmove c1 ([seq ↑[i]_p1 | i <- cs1] ++ cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
last c1 ([seq ↑[i]_p1 | i <- cs1] ++ cs4) = c2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
size ([seq ↑[i]_p1 | i <- cs1] ++ cs4) <=
size cs1 + ((size cs3).+1 + size cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
    -
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
size ([seq ↑[i]_p1 | i <- cs1] ++ cs4) < m
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
path rmove c1 ([seq ↑[i]_p1 | i <- cs1] ++ cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
last c1 ([seq ↑[i]_p1 | i <- cs1] ++ cs4) = c2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
size ([seq ↑[i]_p1 | i <- cs1] ++ cs4) <=
size cs1 + ((size cs3).+1 + size cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4) by rewrite size_cat !size_map (leq_trans _ Lcs) // ltnS;
         rewrite leq_add2l -addSnnS leq_addl.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
path rmove c1 ([seq ↑[i]_p1 | i <- cs1] ++ cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
last c1 ([seq ↑[i]_p1 | i <- cs1] ++ cs4) = c2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
size ([seq ↑[i]_p1 | i <- cs1] ++ cs4) <=
size cs1 + ((size cs3).+1 + size cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
    -
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
path rmove c1 ([seq ↑[i]_p1 | i <- cs1] ++ cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
last c1 ([seq ↑[i]_p1 | i <- cs1] ++ cs4) = c2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
size ([seq ↑[i]_p1 | i <- cs1] ++ cs4) <=
size cs1 + ((size cs3).+1 + size cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4) rewrite cat_path -{1}[c1]cunliftrK path_liftr //=.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
path (move (rrel (n:=3))) ↓[c1] cs1 &&
path rmove (last c1 [seq ↑[i]_p1 | i <- cs1]) cs4
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
last c1 ([seq ↑[i]_p1 | i <- cs1] ++ cs4) = c2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
size ([seq ↑[i]_p1 | i <- cs1] ++ cs4) <=
size cs1 + ((size cs3).+1 + size cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
      rewrite c1'Pcs1.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
true &&
path rmove (last c1 [seq ↑[i]_p1 | i <- cs1]) cs4
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
last c1 ([seq ↑[i]_p1 | i <- cs1] ++ cs4) = c2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
size ([seq ↑[i]_p1 | i <- cs1] ++ cs4) <=
size cs1 + ((size cs3).+1 + size cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
      by rewrite -{1}[c1]cunliftrK last_map lc1'cs1Epp4 
                 -/p1 -p6Ep1 p4Ep2 -p7Ep2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
last c1 ([seq ↑[i]_p1 | i <- cs1] ++ cs4) = c2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
size ([seq ↑[i]_p1 | i <- cs1] ++ cs4) <=
size cs1 + ((size cs3).+1 + size cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
    rewrite last_cat -{1}[c1]cunliftrK last_map.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
last ↑[last ↓[c1] cs1]_(c1 ldisk) cs4 = c2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
size ([seq ↑[i]_p1 | i <- cs1] ++ cs4) <=
size cs1 + ((size cs3).+1 + size cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
    by rewrite lc1'cs1Epp4 -/p1 -p6Ep1 p4Ep2 -p7Ep2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Ep1:p6 = p1
p7Ep2:p7 = p2
size ([seq ↑[i]_p1 | i <- cs1] ++ cs4) <=
size cs1 + ((size cs3).+1 + size cs4)
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
  by rewrite size_cat size_map leq_add2l leq_addl.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
have p6Ep2 : p6 = p2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6 = p2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
  case: (p6 =P p2) => // /eqP p6Dp2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Dp2:p6 != p2
p6 = p2
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
  case/negP:  p5Dp6.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Dp2:p6 != p2
p5 == p6
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
  by rewrite p5Ep opeg3E // eq_sym p6Dp1 eq_sym.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
have p7Ep1 : p7 = p1.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7 = p1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
  apply/eqP.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7 == p1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
  by rewrite opeg3E // p5Ep opeg3E // eqxx.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
minn u v <= size cs1 + ((size cs3).+1 + size cs4)
apply: leq_trans (geq_minr _ _) _.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
v <= size cs1 + ((size cs3).+1 + size cs4)
rewrite -[v]addnA leq_add //.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
size_rpeg ↓[c1] p2 <= size cs1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
2 ^ n + size_rpeg ↓[c2] p1 <= (size cs3).+1 + size cs4
  apply: leq_of_leqif (rpeg_min _ _ ) => //.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
last ↓[c1] cs1 = `c[p2]
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
2 ^ n + size_rpeg ↓[c2] p1 <= (size cs3).+1 + size cs4
  by rewrite lc1'cs1Epp4 p4Ep2.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
2 ^ n + size_rpeg ↓[c2] p1 <= (size cs3).+1 + size cs4
apply: leq_add.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
2 ^ n <= (size cs3).+1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
size_rpeg ↓[c2] p1 <= size cs4
  rewrite -[2 ^ _]prednK ?expn_gt0 // ltnS.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
(2 ^ n).-1 <= size cs3
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
size_rpeg ↓[c2] p1 <= size cs4
  have <-: size_rpeg (↓[c3]) p1 = (2 ^ n).-1.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
size_rpeg ↓[c3] p1 = (2 ^ n).-1
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
size_rpeg ↓[c3] p1 <= size cs3
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
size_rpeg ↓[c2] p1 <= size cs4
    by rewrite cliftrK size_rpegE rpeg_ppeg ?opegDl // size_ppeg.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
size_rpeg ↓[c3] p1 <= size cs3
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
size_rpeg ↓[c2] p1 <= size cs4
  apply: leq_of_leqif (rpeg_min _ _) => //.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
last ↓[c3] cs3 = `c[p1]
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
size_rpeg ↓[c2] p1 <= size cs4
  by rewrite c3'cs3Epp7 p7Ep1.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
size_rpeg ↓[c2] p1 <= size cs4
have [cs4' [c4'Pcs4' lc4'cs4'Elc4cs4 /leq_of_leqif Lcs4']] := 
      pathS_restrict HHr c4Pcs4.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
cs4':seq (configuration 3 n)
c4'Pcs4':path (move (rrel (n:=3))) ↓[c4] cs4'
lc4'cs4'Elc4cs4:last ↓[c4] cs4' = ↓[last c4 cs4]
Lcs4':size cs4' <= size cs4
size_rpeg ↓[c2] p1 <= size cs4
rewrite cliftrK p7Ep1 in c4'Pcs4'.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
cs4':seq (configuration 3 n)
lc4'cs4'Elc4cs4:last ↓[c4] cs4' = ↓[last c4 cs4]
Lcs4':size cs4' <= size cs4
c4'Pcs4':path (move (rrel (n:=3))) `c[p1] cs4'
size_rpeg ↓[c2] p1 <= size cs4
rewrite cliftrK p7Ep1 in lc4'cs4'Elc4cs4.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
cs4':seq (configuration 3 n)
Lcs4':size cs4' <= size cs4
c4'Pcs4':path (move (rrel (n:=3))) `c[p1] cs4'
lc4'cs4'Elc4cs4:last `c[p1] cs4' = ↓[last c4 cs4]
size_rpeg ↓[c2] p1 <= size cs4
apply: leq_trans Lcs4'.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
cs4':seq (configuration 3 n)
c4'Pcs4':path (move (rrel (n:=3))) `c[p1] cs4'
lc4'cs4'Elc4cs4:last `c[p1] cs4' = ↓[last c4 cs4]
size_rpeg ↓[c2] p1 <= size cs4'
apply: leq_of_leqif (lpeg_min _ _) => //.
HHr:irreflexive (rrel (n:=3))
HHs:symmetric (rrel (n:=3))
n:nat
IH:forall (c1 c2 : configuration 3 n)
  (cs : seq (configuration 3 n)),
path rmove c1 cs ->
last c1 cs = c2 ->
size (rhanoi3 c1 c2) <= size cs
c1, c2:configuration 3 n.+1
p:=`p[c1 ldisk, c2 ldisk]:peg 3
p1:=c1 ldisk:(fun=> peg 3) ldisk
p2:=c2 ldisk:(fun=> peg 3) ldisk
p1Dp2:p1 != p2
u:=size_rpeg ↓[c1] p + size_rpeg ↓[c2] p:nat
v:=size_rpeg ↓[c1] p2 + 2 ^ n + size_rpeg ↓[c2] p1:nat
f:=fun_of_fin^~ ldisk:configuration 3 n.+1 -> (fun=> peg 3) ldisk
m:nat
IH1:forall cs : seq (configuration 3 n.+1),
size cs < m ->
path rmove c1 cs ->
last c1 cs = c2 -> minn u v < size cs
cs:seq (configuration 3 n.+1)
cs1:seq (configuration 3 n)
cs2:seq (configuration 3 n.+1)
p1':=c1 ldisk:peg 3
p3:peg 3
p4:=`p[p1', p3]:peg 3
c1':=↓[c1]:configuration 3 n
c3:=↑[`c[p4]]_p3:configuration 3 n.+1
p1Dp3:p1' != p3
p1Rp3:rrel p1' p3
lc1'cs1Epp4:last c1' cs1 = `c[p4]
csE:cs = [seq ↑[i]_p1' | i <- cs1] ++ c3 :: cs2
c1'Pcs1:path (move (rrel (n:=3))) c1' cs1
p1'Ep1:p1' = p1
p3Dp2:p3 != p2
p3Ep:p3 = p
p4Ep2:p4 = p2
cs3:seq (configuration 3 n)
cs4:seq (configuration 3 n.+1)
p5:=c3 ldisk:peg 3
p6:peg 3
p7:=`p[p5, p6]:peg 3
c3':=↓[c3]:configuration 3 n
c4:=↑[`c[p7]]_p6:configuration 3 n.+1
p5Dp6:p5 != p6
p5p6:rrel p5 p6
c3'cs3Epp7:last c3' cs3 = `c[p7]
cs2E:cs2 = [seq ↑[i]_p5 | i <- cs3] ++ c4 :: cs4
c3'Pcs3:path (move (rrel (n:=3))) c3' cs3
c4Pcs4:path (move (rrel (n:=3))) c4 cs4
p5Ep:p5 = p
lc6cs4Ec2:last c4 cs4 = c2
Lcs:size cs1 + (size cs3 + (size cs4).+1) < m
p6Dp1:p6 != p1
p6Ep2:p6 = p2
p7Ep1:p7 = p1
cs4':seq (configuration 3 n)
c4'Pcs4':path (move (rrel (n:=3))) `c[p1] cs4'
lc4'cs4'Elc4cs4:last `c[p1] cs4' = ↓[last c4 cs4]
last `c[p1] cs4' = ↓[c2]
by rewrite lc4'cs4'Elc4cs4 lc6cs4Ec2.
Qed.

Fixpoint size_rhanoi3 {n : nat} : _ _ n -> _ _ n ->  nat :=
  if n is n1.+1 then
      fun c1 c2 : configuration 3 n1.+1 =>
       let p1 := c1 ldisk in
       let p2 := c2 ldisk in
       let c1' := ↓[c1] in
       let c2' := ↓[c2] in
       if p1 == p2 then size_rhanoi3 c1' c2' else
       (* one jump *)
       let p := `p[p1, p2] in
       let m1 := size_rpeg c1' p + size_rpeg c2' p in 
       (* two jumps *)
       let m2 := size_rpeg c1' p2 + 2 ^ n1 + size_rpeg c2' p1
       in (minn m1 m2).+1
  else fun _ _ => 0.

(* size computes the size *)
Lemma size_rhanoi3E n (c1 c2 : _ _ n) : size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2).
n:nat
c1, c2:configuration 3 n
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
Proof.
n:nat
c1, c2:configuration 3 n
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
elim: n c1 c2 => //= n IH c1 c2.
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
(if c1 ldisk == c2 ldisk
 then size_rhanoi3 ↓[c1] ↓[c2]
 else
  (minn
     (size_rpeg ↓[c1] `p[c1 ldisk, c2 ldisk] +
      size_rpeg ↓[c2] `p[c1 ldisk, c2 ldisk])
     (size_rpeg ↓[c1] (c2 ldisk) + 2 ^ n +
      size_rpeg ↓[c2] (c1 ldisk))).+1) =
size
  (if c1 ldisk == c2 ldisk
   then
    [seq ↑[i]_(c1 ldisk) | i <- rhanoi3 ↓[c1] ↓[c2]]
   else
    if
     size_rpeg ↓[c1] `p[c1 ldisk, c2 ldisk] +
     size_rpeg ↓[c2] `p[c1 ldisk, c2 ldisk] <=
     size_rpeg ↓[c1] (c2 ldisk) + 2 ^ n +
     size_rpeg ↓[c2] (c1 ldisk)
    then
     [seq ↑[i]_(c1 ldisk)
        | i <- rpeg ↓[c1] `p[c1 ldisk, c2 ldisk]] ++
     ↑[`c[`p[c1 ldisk, c2 ldisk]]]_(c2 ldisk)
     :: [seq ↑[i]_(c2 ldisk)
           | i <- lpeg `p[c1 ldisk, c2 ldisk] ↓[c2]]
    else
     [seq ↑[i]_(c1 ldisk)
        | i <- rpeg ↓[c1] (c2 ldisk)] ++
     ↑[`c[c2 ldisk]]_(`p[c1 ldisk, c2 ldisk])
     :: [seq ↑[i]_(`p[c1 ldisk, c2 ldisk])
           | i <- ppeg (c2 ldisk) (c1 ldisk)] ++
        ↑[`c[c1 ldisk]]_(c2 ldisk)
        :: [seq ↑[i]_(c2 ldisk)
              | i <- lpeg (c1 ldisk) ↓[c2]])
case: eqP => [E|/eqP NE].
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
E:c1 ldisk = c2 ldisk
size_rhanoi3 ↓[c1] ↓[c2] =
size [seq ↑[i]_(c1 ldisk) | i <- rhanoi3 ↓[c1] ↓[c2]]
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
(minn
   (size_rpeg ↓[c1] `p[c1 ldisk, c2 ldisk] +
    size_rpeg ↓[c2] `p[c1 ldisk, c2 ldisk])
   (size_rpeg ↓[c1] (c2 ldisk) + 2 ^ n +
    size_rpeg ↓[c2] (c1 ldisk))).+1 =
size
  (if
    size_rpeg ↓[c1] `p[c1 ldisk, c2 ldisk] +
    size_rpeg ↓[c2] `p[c1 ldisk, c2 ldisk] <=
    size_rpeg ↓[c1] (c2 ldisk) + 2 ^ n +
    size_rpeg ↓[c2] (c1 ldisk)
   then
    [seq ↑[i]_(c1 ldisk)
       | i <- rpeg ↓[c1] `p[c1 ldisk, c2 ldisk]] ++
    ↑[`c[`p[c1 ldisk, c2 ldisk]]]_(c2 ldisk)
    :: [seq ↑[i]_(c2 ldisk)
          | i <- lpeg `p[c1 ldisk, c2 ldisk] ↓[c2]]
   else
    [seq ↑[i]_(c1 ldisk) | i <- rpeg ↓[c1] (c2 ldisk)] ++
    ↑[`c[c2 ldisk]]_(`p[c1 ldisk, c2 ldisk])
    :: [seq ↑[i]_(`p[c1 ldisk, c2 ldisk])
          | i <- ppeg (c2 ldisk) (c1 ldisk)] ++
       ↑[`c[c1 ldisk]]_(c2 ldisk)
       :: [seq ↑[i]_(c2 ldisk)
             | i <- lpeg (c1 ldisk) ↓[c2]])
  by rewrite size_map; apply: IH.
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
(minn
   (size_rpeg ↓[c1] `p[c1 ldisk, c2 ldisk] +
    size_rpeg ↓[c2] `p[c1 ldisk, c2 ldisk])
   (size_rpeg ↓[c1] (c2 ldisk) + 2 ^ n +
    size_rpeg ↓[c2] (c1 ldisk))).+1 =
size
  (if
    size_rpeg ↓[c1] `p[c1 ldisk, c2 ldisk] +
    size_rpeg ↓[c2] `p[c1 ldisk, c2 ldisk] <=
    size_rpeg ↓[c1] (c2 ldisk) + 2 ^ n +
    size_rpeg ↓[c2] (c1 ldisk)
   then
    [seq ↑[i]_(c1 ldisk)
       | i <- rpeg ↓[c1] `p[c1 ldisk, c2 ldisk]] ++
    ↑[`c[`p[c1 ldisk, c2 ldisk]]]_(c2 ldisk)
    :: [seq ↑[i]_(c2 ldisk)
          | i <- lpeg `p[c1 ldisk, c2 ldisk] ↓[c2]]
   else
    [seq ↑[i]_(c1 ldisk) | i <- rpeg ↓[c1] (c2 ldisk)] ++
    ↑[`c[c2 ldisk]]_(`p[c1 ldisk, c2 ldisk])
    :: [seq ↑[i]_(`p[c1 ldisk, c2 ldisk])
          | i <- ppeg (c2 ldisk) (c1 ldisk)] ++
       ↑[`c[c1 ldisk]]_(c2 ldisk)
       :: [seq ↑[i]_(c2 ldisk)
             | i <- lpeg (c1 ldisk) ↓[c2]])
set p := `p[_, _].
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
p:=`p[c1 ldisk, c2 ldisk]:peg 3
(minn (size_rpeg ↓[c1] p + size_rpeg ↓[c2] p)
   (size_rpeg ↓[c1] (c2 ldisk) + 2 ^ n +
    size_rpeg ↓[c2] (c1 ldisk))).+1 =
size
  (if
    size_rpeg ↓[c1] p + size_rpeg ↓[c2] p <=
    size_rpeg ↓[c1] (c2 ldisk) + 2 ^ n +
    size_rpeg ↓[c2] (c1 ldisk)
   then
    [seq ↑[i]_(c1 ldisk) | i <- rpeg ↓[c1] p] ++
    ↑[`c[p]]_(c2 ldisk)
    :: [seq ↑[i]_(c2 ldisk) | i <- lpeg p ↓[c2]]
   else
    [seq ↑[i]_(c1 ldisk) | i <- rpeg ↓[c1] (c2 ldisk)] ++
    ↑[`c[c2 ldisk]]_p
    :: [seq ↑[i]_p | i <- ppeg (c2 ldisk) (c1 ldisk)] ++
       ↑[`c[c1 ldisk]]_(c2 ldisk)
       :: [seq ↑[i]_(c2 ldisk)
             | i <- lpeg (c1 ldisk) ↓[c2]])
set x := size_rpeg _ _; set y := size_rpeg _ _.
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
p:=`p[c1 ldisk, c2 ldisk]:peg 3
x:=size_rpeg ↓[c1] p:nat
y:=size_rpeg ↓[c2] p:nat
(minn (x + y)
   (size_rpeg ↓[c1] (c2 ldisk) + 2 ^ n +
    size_rpeg ↓[c2] (c1 ldisk))).+1 =
size
  (if
    x + y <=
    size_rpeg ↓[c1] (c2 ldisk) + 2 ^ n +
    size_rpeg ↓[c2] (c1 ldisk)
   then
    [seq ↑[i]_(c1 ldisk) | i <- rpeg ↓[c1] p] ++
    ↑[`c[p]]_(c2 ldisk)
    :: [seq ↑[i]_(c2 ldisk) | i <- lpeg p ↓[c2]]
   else
    [seq ↑[i]_(c1 ldisk) | i <- rpeg ↓[c1] (c2 ldisk)] ++
    ↑[`c[c2 ldisk]]_p
    :: [seq ↑[i]_p | i <- ppeg (c2 ldisk) (c1 ldisk)] ++
       ↑[`c[c1 ldisk]]_(c2 ldisk)
       :: [seq ↑[i]_(c2 ldisk)
             | i <- lpeg (c1 ldisk) ↓[c2]])
set z := size_rpeg _ _; set t := size_rpeg _ _.
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
p:=`p[c1 ldisk, c2 ldisk]:peg 3
x:=size_rpeg ↓[c1] p:nat
y:=size_rpeg ↓[c2] p:nat
z:=size_rpeg ↓[c1] (c2 ldisk):nat
t:=size_rpeg ↓[c2] (c1 ldisk):nat
(minn (x + y) (z + 2 ^ n + t)).+1 =
size
  (if x + y <= z + 2 ^ n + t
   then
    [seq ↑[i]_(c1 ldisk) | i <- rpeg ↓[c1] p] ++
    ↑[`c[p]]_(c2 ldisk)
    :: [seq ↑[i]_(c2 ldisk) | i <- lpeg p ↓[c2]]
   else
    [seq ↑[i]_(c1 ldisk) | i <- rpeg ↓[c1] (c2 ldisk)] ++
    ↑[`c[c2 ldisk]]_p
    :: [seq ↑[i]_p | i <- ppeg (c2 ldisk) (c1 ldisk)] ++
       ↑[`c[c1 ldisk]]_(c2 ldisk)
       :: [seq ↑[i]_(c2 ldisk)
             | i <- lpeg (c1 ldisk) ↓[c2]])
rewrite fun_if.
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
p:=`p[c1 ldisk, c2 ldisk]:peg 3
x:=size_rpeg ↓[c1] p:nat
y:=size_rpeg ↓[c2] p:nat
z:=size_rpeg ↓[c1] (c2 ldisk):nat
t:=size_rpeg ↓[c2] (c1 ldisk):nat
(minn (x + y) (z + 2 ^ n + t)).+1 =
(if x + y <= z + 2 ^ n + t
 then
  size
    ([seq ↑[i]_(c1 ldisk) | i <- rpeg ↓[c1] p] ++
     ↑[`c[p]]_(c2 ldisk)
     :: [seq ↑[i]_(c2 ldisk) | i <- lpeg p ↓[c2]])
 else
  size
    ([seq ↑[i]_(c1 ldisk)
        | i <- rpeg ↓[c1] (c2 ldisk)] ++
     ↑[`c[c2 ldisk]]_p
     :: [seq ↑[i]_p | i <- ppeg (c2 ldisk) (c1 ldisk)] ++
        ↑[`c[c1 ldisk]]_(c2 ldisk)
        :: [seq ↑[i]_(c2 ldisk)
              | i <- lpeg (c1 ldisk) ↓[c2]]))
rewrite size_cat /= size_cat /= size_cat /= !size_map.
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
p:=`p[c1 ldisk, c2 ldisk]:peg 3
x:=size_rpeg ↓[c1] p:nat
y:=size_rpeg ↓[c2] p:nat
z:=size_rpeg ↓[c1] (c2 ldisk):nat
t:=size_rpeg ↓[c2] (c1 ldisk):nat
(minn (x + y) (z + 2 ^ n + t)).+1 =
(if x + y <= z + 2 ^ n + t
 then size (rpeg ↓[c1] p) + (size (lpeg p ↓[c2])).+1
 else
  size (rpeg ↓[c1] (c2 ldisk)) +
  (size (ppeg (c2 ldisk) (c1 ldisk)) +
   (size (lpeg (c1 ldisk) ↓[c2])).+1).+1)
rewrite -!(size_rpegE, size_lpegE) /=.
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
p:=`p[c1 ldisk, c2 ldisk]:peg 3
x:=size_rpeg ↓[c1] p:nat
y:=size_rpeg ↓[c2] p:nat
z:=size_rpeg ↓[c1] (c2 ldisk):nat
t:=size_rpeg ↓[c2] (c1 ldisk):nat
(minn (x + y) (z + 2 ^ n + t)).+1 =
(if x + y <= z + 2 ^ n + t
 then size_rpeg ↓[c1] p + (size_rpeg ↓[c2] p).+1
 else
  size_rpeg ↓[c1] (c2 ldisk) +
  (size (ppeg (c2 ldisk) (c1 ldisk)) +
   (size_rpeg ↓[c2] (c1 ldisk)).+1).+1)
rewrite -/x -/y -/z -/t.
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
p:=`p[c1 ldisk, c2 ldisk]:peg 3
x:=size_rpeg ↓[c1] p:nat
y:=size_rpeg ↓[c2] p:nat
z:=size_rpeg ↓[c1] (c2 ldisk):nat
t:=size_rpeg ↓[c2] (c1 ldisk):nat
(minn (x + y) (z + 2 ^ n + t)).+1 =
(if x + y <= z + 2 ^ n + t
 then x + y.+1
 else
  z + (size (ppeg (c2 ldisk) (c1 ldisk)) + t.+1).+1)
rewrite size_ppeg -[_ + t.+1]addSnnS prednK ?expn_gt0 //.
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
p:=`p[c1 ldisk, c2 ldisk]:peg 3
x:=size_rpeg ↓[c1] p:nat
y:=size_rpeg ↓[c2] p:nat
z:=size_rpeg ↓[c1] (c2 ldisk):nat
t:=size_rpeg ↓[c2] (c1 ldisk):nat
(minn (x + y) (z + 2 ^ n + t)).+1 =
(if x + y <= z + 2 ^ n + t
 then x + y.+1
 else z + (2 ^ n + t).+1)
rewrite !addnS.
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
p:=`p[c1 ldisk, c2 ldisk]:peg 3
x:=size_rpeg ↓[c1] p:nat
y:=size_rpeg ↓[c2] p:nat
z:=size_rpeg ↓[c1] (c2 ldisk):nat
t:=size_rpeg ↓[c2] (c1 ldisk):nat
(minn (x + y) (z + 2 ^ n + t)).+1 =
(if x + y <= z + 2 ^ n + t
 then (x + y).+1
 else (z + (2 ^ n + t)).+1)
rewrite /minn !addnA.
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
p:=`p[c1 ldisk, c2 ldisk]:peg 3
x:=size_rpeg ↓[c1] p:nat
y:=size_rpeg ↓[c2] p:nat
z:=size_rpeg ↓[c1] (c2 ldisk):nat
t:=size_rpeg ↓[c2] (c1 ldisk):nat
(if x + y < z + 2 ^ n + t
 then x + y
 else z + 2 ^ n + t).+1 =
(if x + y <= z + 2 ^ n + t
 then (x + y).+1
 else (z + 2 ^ n + t).+1)
case: leqP => LL1; case: leqP => LL2 //.
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
p:=`p[c1 ldisk, c2 ldisk]:peg 3
x:=size_rpeg ↓[c1] p:nat
y:=size_rpeg ↓[c2] p:nat
z:=size_rpeg ↓[c1] (c2 ldisk):nat
t:=size_rpeg ↓[c2] (c1 ldisk):nat
LL1:z + 2 ^ n + t <= x + y
LL2:x + y <= z + 2 ^ n + t
(z + 2 ^ n + t).+1 = (x + y).+1
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
p:=`p[c1 ldisk, c2 ldisk]:peg 3
x:=size_rpeg ↓[c1] p:nat
y:=size_rpeg ↓[c2] p:nat
z:=size_rpeg ↓[c1] (c2 ldisk):nat
t:=size_rpeg ↓[c2] (c1 ldisk):nat
LL1:x + y < z + 2 ^ n + t
LL2:z + 2 ^ n + t < x + y
(x + y).+1 = (z + 2 ^ n + t).+1
  by congr (_.+1); apply/eqP; rewrite eqn_leq LL1.
n:nat
IH:forall c1 c2 : configuration 3 n,
size_rhanoi3 c1 c2 = size (rhanoi3 c1 c2)
c1, c2:configuration 3 n.+1
NE:c1 ldisk != c2 ldisk
p:=`p[c1 ldisk, c2 ldisk]:peg 3
x:=size_rpeg ↓[c1] p:nat
y:=size_rpeg ↓[c2] p:nat
z:=size_rpeg ↓[c1] (c2 ldisk):nat
t:=size_rpeg ↓[c2] (c1 ldisk):nat
LL1:x + y < z + 2 ^ n + t
LL2:z + 2 ^ n + t < x + y
(x + y).+1 = (z + 2 ^ n + t).+1
by congr (_.+1); apply/eqP; rewrite eqn_leq ltnW // ltnW.
Qed.

Lemma gdist_rhanoi3 n (c1 c2 : _ _ n) : `d[c1, c2]_rmove = size_rhanoi3 c1 c2.
n:nat
c1, c2:configuration 3 n
`d[c1, c2]_rmove = size_rhanoi3 c1 c2
Proof.
n:nat
c1, c2:configuration 3 n
`d[c1, c2]_rmove = size_rhanoi3 c1 c2
apply/eqP; rewrite eqn_leq.
n:nat
c1, c2:configuration 3 n
`d[c1, c2]_rmove <= size_rhanoi3 c1 c2 <=
`d[c1, c2]_rmove
rewrite [size_rhanoi3 _ _]size_rhanoi3E gdist_path_le //=; last 2 first.
n:nat
c1, c2:configuration 3 n
path rmove c1 (rhanoi3 c1 c2)
n:nat
c1, c2:configuration 3 n
last c1 (rhanoi3 c1 c2) = c2
n:nat
c1, c2:configuration 3 n
size (rhanoi3 c1 c2) <= `d[c1, c2]_rmove
-
n:nat
c1, c2:configuration 3 n
path rmove c1 (rhanoi3 c1 c2)
n:nat
c1, c2:configuration 3 n
last c1 (rhanoi3 c1 c2) = c2
n:nat
c1, c2:configuration 3 n
size (rhanoi3 c1 c2) <= `d[c1, c2]_rmove by apply: path_rhanoi3.
n:nat
c1, c2:configuration 3 n
last c1 (rhanoi3 c1 c2) = c2
n:nat
c1, c2:configuration 3 n
size (rhanoi3 c1 c2) <= `d[c1, c2]_rmove
-
n:nat
c1, c2:configuration 3 n
last c1 (rhanoi3 c1 c2) = c2
n:nat
c1, c2:configuration 3 n
size (rhanoi3 c1 c2) <= `d[c1, c2]_rmove by apply: last_rhanoi3.
n:nat
c1, c2:configuration 3 n
size (rhanoi3 c1 c2) <= `d[c1, c2]_rmove
have /gpath_connect [p1 p1H] : connect rmove c1 c2 by apply:  move_connect.
n:nat
c1, c2:configuration 3 n
p1:seq
  (finfun_finType (ordinal_finType n)
     (ordinal_finType 3))
p1H:gpath rmove c1 c2 p1
size (rhanoi3 c1 c2) <= `d[c1, c2]_rmove
rewrite (gpath_dist p1H) rhanoi3_min //; first apply: gpath_path p1H.
n:nat
c1, c2:configuration 3 n
p1:seq
  (finfun_finType (ordinal_finType n)
     (ordinal_finType 3))
p1H:gpath rmove c1 c2 p1
last c1 p1 = c2
by apply: gpath_last p1H.
Qed.

End Hanoi3.